Positive density dependence acting on mortality can help maintain species-rich communities

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Version of Record published: June 18, 2020 (This version)
Accepted: May 22, 2020
Received: April 22, 2020

1. Of interest
Gut Health: The value of connections

Vanessa Rossetto Marcelino

Insight Sep 19, 2023
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Abstract

Conspecific negative density dependence is ubiquitous and has long been recognized as an important factor favoring the coexistence of competing species at local scale. By contrast, a positive density-dependent growth rate is thought to favor species exclusion by inhibiting the growth of less competitive species. Yet, such conspecific positive density dependence often reduces extrinsic mortality (e.g. reduced predation), which favors species exclusion in the first place. Here, using a combination of analytical derivations and numerical simulations, I show that this form of positive density dependence can favor the existence of equilibrium points characterized by species coexistence. Those equilibria are not globally stable, but allow the maintenance of species-rich communities in multispecies simulations. Therefore, conspecific positive density dependence does not necessarily favor species exclusion. On the contrary, some forms of conspecific positive density dependence may even help maintain species richness in natural communities. These results should stimulate further investigations into the precise mechanisms underlying density dependence.

Introduction

The tremendous diversity of species in ecological communities has long motivated ecologists to explore how this diversity is maintained (Hutchinson, 1959; Chesson, 2000; Hubbell, 2001; Levine et al., 2017). Species richness in a local community is the result of several processes that act at different scales, none of them being mutually exclusive (Götzenberger et al., 2012). At regional and global scales, these include randomness and dispersal processes (Hubbell, 2001; Leibold and Chase, 2018). At local scale, in addition to abiotic factors (physical constraints of the environment), biotic interactions determine community assembly (Gause, 1934; Hutchinson, 1961; Holt, 1977; Tilman, 1982; Chesson, 2000; Gross, 2008). In particular, some species are better competitors than others, and competitive imbalances can lead to the exclusion of less competitive species (Gause, 1934; Levin, 1970; Tilman, 1982; Meszéna et al., 2006).

Interspecific competition for resources (see Box 1 for a glossary of terms in italics) has been recognized as one of the main drivers of species exclusion (Gause, 1934; Tilman, 1982). Additionally, interspecific reproductive interference – i.e. any interspecific sexual interaction reducing the reproductive success of females – can inhibit species coexistence. Such interspecific sexual interactions are common in nature, especially among closely related species (Gröning and Hochkirch, 2008), and can cause species exclusion more easily than competition for resources, as shown theoretically (Kuno, 1992; Yoshimura and Clark, 1994; Kishi and Nakazawa, 2013; Schreiber et al., 2019) and empirically in some species (Takafuji et al., 1997; Kishi et al., 2009; Takakura et al., 2009; Crowder et al., 2010; Crowder et al., 2011).

Box 1.

Glossary.

Interactions
- Intraspecific competition for resources – Any form of competition in which conspecifics (i.e. individuals of the same species) compete for resources.
- Interspecific competition for resources – Any form of competition in which heterospecifics (i.e. individuals belonging to different species) compete for resources.
- Interspecific reproductive interference – Any sexual interaction in which heterospecifics reduce female reproductive success (or female function).
Density dependence
- Conspecific negative density dependence – Decline in the population growth rate with increasing local density of conspecifics. It typically results from intraspecific competition for resources.
- Heterospecific negative density dependence – Decline in the population growth rate with increasing local density of heterospecifics. It typically results from interspecific competition for resources.
- Conspecific positive density dependence – Increase in the population growth rate with increasing local density of conspecifics. It can arise from many different mechanisms (Figure 1). This is the focus of this study.
- Heterospecific positive density dependence – Increase in the population growth rate with increasing local density of heterospecifics. It can arise from any mutualistic interaction (e.g. interspecific facilitation among plant species). This is not the focus of this study.
Frequency dependence
- Heterospecific negative frequency dependence – Decline in the population growth rate with increasing local frequency of heterospecifics vs. conspecifics. It typically results from interspecific reproductive interference. This can also be called conspecific positive frequency dependence.
- Heterospecific positive frequency dependence – Increase in the population growth rate with increasing local frequency of heterospecifics vs. conspecifics. It typically results from the interplay between intraspecific and interspecific competition for resources. This can also be called conspecific negative frequency dependence.

In the face of these negative interspecific interactions, many mechanisms favoring species coexistence have been identified. These include niche separation (Gause, 1934), predatory/herbivory interactions (Chesson and Huntly, 1997), positive interactions (Gross, 2008), crowding effects (Gavina et al., 2018), and individual-level variations (Uriarte and Menge, 2018; but see Hart et al., 2016). Notably, species often differ in their use of multiple-limiting resources (Tilman, 1982), causing species to limit their own population growths more than they limit others (Adler et al., 2007). A core tenet of Chesson’s coexistence theory – one of the well-developed coexistence theories – is precisely that negative density dependence must be stronger among conspecifics (conspecific negative density dependence) than among heterospecifics (heterospecific negative density dependence) for species to coexist in two-species systems (Chesson, 2000). Even though this criterion does not hold in multispecies communities (Barabás et al., 2016; Song et al., 2019), the importance of conspecific negative density dependence – when the growth rate of a population decreases as its density increases – for species coexistence in two-species systems is well accepted (MacArthur, 1970; Chesson, 2000; McPeek, 2012). Indeed, conspecific negative density dependence favors the existence of a coexistence equilibrium (i.e. the ‘feasibility condition’ is fulfilled) that is globally stable (all species can invade even if they are rare initially; i.e. the ‘global stability condition’ is fulfilled) (Case, 2000).

In many species, however, individuals benefit from the presence of conspecifics, resulting in conspecific positive density dependence – i.e. the growth rate of a population increases as its density increases (a phenomenon that is commonly referred to as ‘Allee effect’ when it occurs at low density). Contrary to conspecific negative density dependence, conspecific positive density dependence can inhibit the coexistence of species interacting negatively with each other (e.g. via competition for resources or reproductive interference) by reducing even further the growth rate of inferior competitors that are at lower density than superior competitors (as shown theoretically by Wang et al., 1999 and De Silva and Jang, 2015). More precisely, the effect of conspecific positive density dependence on coexistence is two-fold. First, it can constrain the conditions under which there is stable coexistence (i.e. the ‘feasibility condition’ is constrained). Second, it can inhibit the invasion of species that are at low density initially (i.e. the ‘global stability condition’ is unfulfilled). Therefore, in concert with competition for resources and reproductive interference, conspecific positive density dependence can be a potent mechanism inhibiting the coexistence of competing species. Note that heterospecific positive density dependence – when the growth rate of a population increases as the density of heterospecifics increases – has also been documented (it can arise from any mutualistic interaction; for example Bruno et al., 2003) and can promote species coexistence (Gross, 2008). In the present study, however, I focus exclusively on the implication of conspecific positive density dependence for coexistence.

Conspecific positive density dependence has been described for most major animal taxa (reviewed by Courchamp et al., 1999; Stephens et al., 1999; Kramer et al., 2009), and can be caused by a variety of mechanisms, such as mate limitation (Gascoigne et al., 2009), cooperative feeding (Carbone et al., 1997), cooperative defense (Angulo et al., 2018), predator satiation (Gascoigne and Lipcius, 2004) or anti-predator strategies (like aposematism, Mallet and Joron, 1999; Joron and Iwasa, 2005). In plants, conspecific positive density dependence can be driven by inbreeding depression (Willi et al., 2005), pollen limitation (Sih and Baltus, 1987; Groom, 1998) or substrate modification (favoring seedling establishment, Bruno et al., 2003). Interestingly, conspecific positive dependence can associate with increased reproduction or reduced mortality (Figure 1; as emphasized by Berec et al., 2007). However, theoretical models investigating the determinants of species coexistence have considered conspecific positive density dependence acting on birth rates (Wang et al., 1999) or net growth rates (defined as the difference between birth rate and mortality rate; De Silva and Jang, 2015), but not on mortality rates alone. Decreased mortality rates at high population density give rise to conspecific positive density dependence in many species (Figure 1; and see the description of specific natural history examples in Figure 2), yet the effect of this form of conspecific positive density dependence on species coexistence has not been investigated explicitly.

Figure 1

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Mechanisms causing conspecific positive density dependence, and affecting reproduction or mortality.

For references and for a more exhaustive list, see the reviews by Courchamp et al., 1999; Stephens et al., 1999; Berec et al., 2007; Kramer et al., 2009. Note that two or more mechanisms causing positive or negative density dependence can occur simultaneously (Berec et al., 2007).

Figure 2

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Example of organisms undergoing conspecific positive density dependence that associates with reduced mortality.

(a) Heliconius butterflies are aposematic (toxic and conspicuous) and therefore benefit from reduced predation when conspecifics are abundant. (b) Black-browed albatrosses in large flocks benefits from increased foraging efficiency and therefore from increased survival (Grünbaum and Veit, 2003). (c) Bottlebrush squirreltail in high density have a high survival rate and a high establishement success in arid grasslands (Sheley and James, 2014). (d) Meerkats in large group benefit from increased vigilance at the group level, reducing the risk of predation (Clutton-Brock et al., 1999). (e) Wild dogs in large group benefit from a high hunting success and defend effectively their kill against kleptoparasitism, thereby increasing their survival (Fanshawe and Fitzgibbon, 1993; Carbone et al., 1997). (f) Ribbed mussels in high density benefit from reduced crab predation and from improved winter ice resistance (Bertness and Grosholz, 1985). Photo credits: (a) Ettore Balocchi; (b) Ed Dunens; (c) Jeffry B. Mitton; (d) Ronnie Macdonald; (e) Barbara Evans; (f) Kerry Wixted.

One might expect that conspecific positive density dependence should inhibit species coexistence whatever the exact underlying mechanism; in all cases, the most competitive species is at high density and therefore benefits the most from conspecific positive density dependence, thereby promoting species exclusion (Wang et al., 1999; De Silva and Jang, 2015). Nonetheless, density dependence acting on extrinsic mortality (e.g. density-dependent predation) also changes the overall magnitude of extrinsic mortality, yet extrinsic mortality can inhibit species coexistence in the first place (as shown by Holt, 1985, using the global stability criterion). For this reason, conspecific positive density dependence acting only on mortality may yield to a different outcome in term of species coexistence.

The primary goal of this analysis is to show that conspecific positive density dependence associated with reduced mortality can help maintain coexistence among competing species. First, I analyze analytically a two-species model with asymmetric resource competition and conspecific positive density dependence acting on mortality, and I show that a locally stable coexistence equilibrium often exists under such positive density dependence. Second, using numerically simulations, I show that this form of conspecific positive density dependence substantially increases the coexistence region in models accounting for other forms of species asymmetry, as differences in basal mortality (in addition to symmetric resource competition) or asymmetric reproductive interference. In other words, this form of conspecific positive density dependence generally inhibits the global stability of coexistence but can also increase the feasibility domain of the coexistence equilibrium. Third, using simulations of a multi- (> 2) species model, I then show that conspecific positive density dependence associated with reduced mortality can thereby maintain species-rich communities.

Model

Two-species model with asymmetric competition for resources

Model

First, I analyze a simple model accounting for (1) asymmetry in competitive abilities among species and (2) a positive density-dependent mortality term. By being analytically tractable, this model precisely pinpoints the effect of conspecific positive density dependence associated with reduced mortality on species coexistence – i.e. on the feasibility and the global stability of the coexistence equilibrium. All analytical derivations and formal mathematical justifications for the results in this section are detailed in Supplementary files 1A and 1B. The computer code of the simulations and of the analyses is provided as Source code 1 (Python, version 2.7.15).

I consider Lotka-Volterra competition equations, with rescaled variables $t$ , $n_{1}$ and $n_{2}$ that are dimensionless. Namely, $t$ is time scaled to the growth rate, and $n_{1}$ and $n_{2}$ denote the abundances of the two competing species relative to their carrying capacity (scaling is detailed in Supplementary file 1A, and follows Joron and Iwasa, 2005). Both species are assumed to exhibit conspecific negative density dependence due to competition for resources, such that changes in abundances are chosen as logistic regulation rules. Additionally, the presence of species 1 inhibits the growth of species 2, whereas species 2 has no effect on the growth of species 1 – i.e. there is asymmetric interspecific competition for resources leading to heterospecific negative density dependence in one species only. In addition to local competition for resources, conspecific positive density dependence acting on a mortality term affects species abundances. For instance, this additional term of mortality may approximate the direct effect of predation upon the set of competing species (as assumed by Holt, 1985, and Joron and Iwasa, 2005). Aside from asymmetric competition, species are assumed to be similar in order to keep the model analytically tractable. In particular, species have the same carrying capacity, and the growth of both species is constrained by the same type of conspecific density dependence (same conspecific negative density dependence driven by competition for resources, and same conspecific positive density dependence acting on mortality). Dynamics are governed by the equations:

{\begin{cases} \frac{d n_{1}}{d t} & = n_{1} [1 - n_{1} - d \times D (n_{1})] \\ \frac{d n_{2}}{d t} & = n_{2} [1 - n_{2} - α_{1} n_{1} - d \times D (n_{2})] \end{cases}

With the non-linear density-dependent mortality function:

D (n_{i}) = \frac{1}{1 + s n_{i}}

Parameter $α_{1} \in [0, 1]$ reflects the intensity of asymmetric competition for resources. If $α_{1} = 0$ , the population dynamics of the two species are independent from each other. If $α_{1} > 0$ , species 1 benefits from a competitive advantage and inhibits the growth of species 2 (heterospecific negative density dependence). Additionally, negative density dependence is assumed to be stronger among conspecifics than among heterospecifics, hence $α_{1} \leq 1$ (following the general coexistence criterion in a two-species system; Chesson, 2000). The density-dependence factor $s \geq 0$ corresponds to the decline rate of mortality at $n_{i} = 0$ ( $D^{'} (0) = - s$ ) (Mallet and Joron, 1999; Joron and Iwasa, 2005). If $s = 0$ , the two species have the same basal density-independent mortality rate $d \in [0, 1]$ (relative to the growth rate); this corresponds to the situation without positive density dependence in the system. If $s > 0$ , density-dependent mortality is characterized by a hyperbolic decrease with species abundance – the most abundant species has the lowest mortality rate $d \times D (n_{i})$ (Figure 3). In supplementary analyses, other linear or non-linear mortality functions are implemented (Figure 3—figure supplement 1).

Figure 3 with 3 supplements see all

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Non-linear mortality functions with different density-dependence factors ( $s$ ).

Pairs of arrows illustrate the non-linear effect of increased $s$ on mortality reduction. Depending on the value of the density-dependence factor $s$ , the most abundant species either benefits the most (red arrows) or the least (blue arrows) from increased $s$ .

For $s > 10^{2}$ , mortality is strongly reduced, even at very low density, and there is almost no reduction of mortality over most of the range of density. This extreme situation pinpoints the effects of increased $s$ , and the full range of $s > 0$ is therefore considered. Remember however that there is a reduction of mortality with increased density at all densities – i.e. conspecific positive density dependence occurs at all densities – only for $s \in [0, 10^{2}]$ (as emphasized in all graphs).

The ‘usual’ form of the Lotka-Volterra model from Equation 1a can easily be derived (following Williams and Banyikwa, 1981). The density-dependent mortality term decreases not only effective intrinsic growth rates, but also effective carrying capacities – i.e. density values reached at equilibrium without competitor (see Supplementary file 1A). In particular, increased density-dependence factor ( $s$ ) associates with high effective carrying capacities, by reducing the intensity of the density-dependent mortality term. As a result, the positive density-dependence factor $s$ is likely to determine the nature of the equilibrium points of the system.

Note that the population dynamic of species 1 is not affected by the density of species 2; only species 2 can be excluded.

Analytical resolution

Depending on the parameters considered, this system can be characterized by a single stable equilibrium point where both species coexist (i.e. a ‘feasible’ equilibrium point of coexistence where both species persist with $n_{i} > 0$ , hereafter called ‘coexistence equilibrium’) (white and gray zones in Figure 4a–c; e.g., Figure 5a and Figure 5c). In particular, under strong asymmetric competition, conspecific positive density dependence characterized by an intermediate factor ( $s$ ) leads to the loss of the coexistence equilibrium (black zone in Figure 4; e.g., Figure 5b). Above a threshold value (identified analytically in Supplementary file 1B, and represented by a dashed red line in Figure 4), increased density-dependence factor $s$ therefore favors the existence of the coexistence equilibrium.

Figure 4 with 4 supplements see all

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Effects of asymmetric competition for resources ( $α_{1}$ ) and positive density-dependence factor ( $s$ ) on coexistence.

Different values of the basal mortality rate ( $d$ ) are also tested. If the coexistence equilibrium exists, it is either a global attractor (it is reached independently of the initial conditions, as long as the two species start at density $> 0$ ) or a local attractor (it is not reached if the least competitive species is initially at a low density) (**a–c**). In each subfigure, the range of $s$ is arbitrarily divided in two; one range where there is conspecific positive density dependence at all densities, and another range where positive density-dependent mortality is very low even at very low densities (one might not consider those cases as conspecific positive density dependence). The dashed red lines correspond to the $s$ values above which increased density-dependence factor ( $s$ ) can favor coexistence (see Supplementary file 1B). Note the use of a logarithmic y-scale; the dashed red lines are not planes of symmetry. Orange dots corresponds to the combinations of parameters tested in Figure 5. Densities of the least competitive species at coexistence equilibrium are also represented (**d–f**). The growth rate of the most competitive species is not affected by the other species. Therefore, at the coexistence equilibrium, the density of the most competitive species correspond to the density of the least competitive species if $α_{1} = 0$ .

Figure 5

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Effect of the positive density-dependence factor ( $s$ ) on the zero-net-growth isoclines in the case of asymmetric competition for resources.

Gray arrows represent the directions of the deterministic changes of species densities. Red and blue lines correspond to the isoclines (when $d n_{i} / d t = 0$ ) for species 1 and 2, respectively. Black and gray points represent stable and unstable equilibria, respectively. In each panel, the nature of the stable equilibria is annotated. Panel (a) corresponds to a case with a coexistence equilibrium that is a global attractor, panel (b) to a case without coexistence equilibrium, and panel (c) to a case with a coexistence equilibrium that is a local attractor. Other parameters: $d = 0.9$ and $α_{1} = 0.5$ , corresponding to the combinations of parameters represented by orange dots in Figure 4c and Figure 4f.

The coexistence equilibrium is always locally stable when it exists (Figure 5). Moreover, for a low density-dependence factor $s$ , the least competitive species can invade when it is rare, just like without conspecific positive density dependence (when $s = 0$ ; Holt, 1985). In other words, the coexistence equilibrium fulfills the condition of global stability – i.e. the coexistence equilibrium is a global attractor (white zone in Figure 4a–c; e.g., Figure 5a). For a high density-dependence factor $s$ , however, the least competitive species cannot invade. The coexistence equilibrium does not fulfill the condition of global stability – i.e. the coexistence equilibrium is a local attractor (gray zone in Figure 4a–c; e.g., Figure 5c). Therefore, increased density-dependence factor $s$ can favor the existence of a coexistence equilibrium that is not attained if the least competitive species is initially too rare – i.e. it can increase the feasibility of coexistence, despite inhibiting global stability.

At coexistence equilibrium, the least competitive species is always at lower density than the most competitive species (Figure 4d–f). Therefore, one might expect that asymmetric competition and density-dependent mortality should act synergistically and complementarily to promote competitive exclusion, thereby inhibiting the feasibility of coexistence. Yet, this model predicts a non-linear effect of the density-dependence factor ( $s$ ) on the feasibility of coexistence. A feasible coexistence equilibrium arises for high density-dependence factors. The reason is simple: species at high density do not necessarily benefit the most from increased density-dependence factor (see pairs of arrows representing changes in mortality with increased $s$ , in Figure 3). For example, in the case of extreme density-dependence factors, species at high density do not benefit from increased density-dependence factor because they do not incur density-dependent mortality (e.g. $D (1) ≃ 0$ for $s = 10^{4}$ ). Mathematically, this effect can be made clear by considering the non-linearity of the partial derivative $\frac{\partial D}{\partial s}$ as a function of $n_{i}$ ; increased density-dependence factor only slightly decreases the mortality rate of species at high density ( $\frac{\partial D}{\partial s}$ close to 0 for high $n_{i}$ , Figure 3—figure supplement 2 and Figure 3—figure supplement 3). Now, suppose that coexistence occurs, with the least competitive species being at lower density than the most competitive species. In that case, by greatly decreasing the mortality rate of the least abundant species, increased density-dependence factor $s$ increases the density of the least competitive species (Figure 4d–f and Figure 4—figure supplement 1), and can therefore facilitate its maintenance. In conditions where the least competitive species is excluded, this same effect can promote the feasibility of coexistence.

This phenomenon is not specific to extreme cases of conspecific positive density dependence; increased density-dependence factor can increase the feasibility of coexistence among competing species even for intermediate density-dependence factors (dashed red line at $s < 10^{2}$ in Figure 4), where there is still a non-linear positive gain due to high density (i.e. saturation does not occur at very low density as for $s = 10^{4}$ , see Figure 3). Notably, this phenomenon occurs when other non-linear or linear mortality functions are implemented (using a numerical method validated in Figure 4—figure supplement 2; Figure 4—figure supplement 3 and Figure 4—figure supplement 4).

Overall, conspecific positive density dependence characterized by a high factor $s$ leads to coexistence that is a local attractor, which is often considered a ‘weaker’ form of coexistence, because it cannot be assembled easily through invasion, and because it is not robust to stochasticity and strong perturbation (Chesson, 2000). Additionally, conspecific positive density dependence characterized by an intermediate factor $s$ strongly inhibits the feasibility of coexistence (which is globally stable without positive density dependence, when $s$ is close to 0). In this model with asymmetric competition for resources, conspecific positive density dependence is therefore best seen as a mechanism inhibiting coexistence. Yet, as we shall see in the next section, conspecific positive density dependence associated with reduced mortality can substantially increase the feasibility domain of the coexistence equilibrium when competitive exclusion is driven by other forms of species asymmetry. This relies on the same effect of positive density dependence on coexistence than the one characterized analytically above.

Two-species models with other forms of competitive exclusion

Models

I now test whether conspecific positive density dependence acting on mortality increases the feasibility of coexistence in models accounting for other forms of species asymmetry. Each of those models include (1) symmetric competition for resources, (2) asymmetry among species, and (3) a positive density-dependent mortality term. Asymmetry among species takes the form of either differences in basal mortality rates (Equation 2) or asymmetric reproductive interference (Equation 3). All other assumptions are the same as in the model with asymmetric competition for resources analyzed above. These systems of equations include additional linear and nonlinear terms, making analytical resolutions difficult. The conditions of existence of the coexistence equilibrium as a local or global attractor are therefore assessed using a numerical method (full methods appear in Supplementary file 1C). This method is validated by applying it to the model described in the previous section and by comparing the results to the analytical ones (Figure 4—figure supplement 2).

In the model with differences in basal mortality rates, population dynamics are governed by the equations:

{\begin{cases} \frac{d n_{1}}{d t} & = n_{1} [1 - n_{1} - α n_{2} - d \times D (n_{1})] \\ \frac{d n_{2}}{d t} & = n_{2} [1 - n_{2} - α n_{1} - [d + δ (1 - d)] \times D (n_{2})] \end{cases}

Parameter $α \in [0, 1]$ corresponds to the intensity of symmetric competition for resources (heterospecific negative density dependence). Parameter $δ \in [0, 1]$ represents the difference in basal mortality rate between species. If $δ = 0$ , the two species have the same basal mortality rate $d$ . If $δ > 0$ , species 2 incurs a higher basal mortality rate than species 1. Therefore, $δ$ reflects the intensity of species asymmetry, just like parameter $α_{1}$ in Equation 1a. The positive density-dependent mortality function $D (n_{i})$ is modeled as in Equation 1b and is characterized by its density-dependence factor $s$ .

In the model with asymmetric reproductive interference, population dynamics are governed by the equations:

{\begin{cases} \frac{d n_{1}}{d t} & = n_{1} [1 - n_{1} - α n_{2} - d \times D (n_{1})] \\ \frac{d n_{2}}{d t} & = n_{2} [\frac{1}{1 + α_{1}^{'} \frac{n_{1}}{n_{2}}} - n_{2} - α n_{1} - d \times D (n_{2})] \end{cases}

Here, parameter $α_{1}^{'} \in [0, 1]$ reflects the intensity of asymmetric reproductive interference (Yoshimura and Clark, 1994; Kishi and Nakazawa, 2013), assuming that sexual interactions are stronger among conspecifics than among heterospecifics, hence $α_{1}^{'} \leq 1$ . If $α_{1}^{'} = 0$ , there is no reproductive interference. If $α_{1}^{'} > 0$ , interspecific sexual interactions reduce the reproductive success of females of species 2 – i.e. the presence of species 1 inhibits the growth rate of species 2 such that $\frac{1}{1 + α_{1}^{'} \frac{n_{1}}{n_{2}}} < 1$ . Contrary to competition for resources that is density-dependent, reproductive interference is frequency-dependent (heterospecific negative frequency dependence); the presence of few heterospecifics might substantially decrease reproductive success as long as they are more abundant than conspecifics, ultimately leading to species exclusion (hence the term $\frac{n_{1}}{n_{2}}$ in the denominator, following Kishi and Nakazawa, 2013).

Numerical resolutions

Depending on the parameters considered, those systems can be characterized by a single stable coexistence equilibrium (white and gray zones in Figure 6). Depending on the form of species asymmetry, the feasibility domain of species coexistence is more or less constrained. In particular, reproductive interference is more prone to species exclusion than the other forms of species asymmetry tested (e.g. see comparable graphs in Figure 6—figure supplement 1 vs. Figure 6—figure supplement 2). Likewise, under asymmetric reproductive interference which is frequency-dependent, coexistence is only ever a local attractor; the least competitive species cannot invade if it is too rare initially and the condition of global stability is never fulfilled (as shown by Kishi and Nakazawa, 2013). Importantly, in those two models, the least competitive species can be excluded when there is no positive density dependence (when $s$ is close to 0, Figure 6), contrary to the case with asymmetric competition for resources. This is not surprising: compared to Equation 1a, the expression of $d n_{2} / d t$ is characterized by an additional negative term ( $- n_{2} \times δ (1 - d) \times D (n_{2})$ ) in Equation 2 or by a low intrinsic growth rate ( $n_{2} \times \frac{1}{1 + α_{1}^{'} \frac{n_{1}}{n_{2}}} \leq n_{2}$ ) in Equation 3. Those additional terms governing population dynamics favor the exclusion of the least competitive species, despite the reduced growth rate of the most competitive species (term $n_{1} \times (- α n_{2})$ in Equations 2 and 3.

Figure 6 with 6 supplements see all

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Effects of difference in basal mortality between species ( $δ$ ) (**a,b,c**) or asymmetric reproductive interference ( $α_{1}^{'}$ ) (**d,e,f**), and positive density-dependence factor ( $s$ ) on coexistence, using the numerical resolution method (systems of Equations 2 and 3, respectively).

If the coexistence equilibrium exists, it is either a global attractor or a local attractor. In each subfigure, the range of $s$ is arbitrarily divided in two; one range where there is conspecific positive density dependence at all densities, and another range where positive density-dependent mortality is very low even at very low densities. Because those two forms of species asymmetry do not promote species exclusion to the same extent, the levels of symmetric competition for resources are chosen as: $α = 0.5$ (**a,b,c**) or $α = 0$ (**d,e,f**).

For asymmetric basal mortality and asymmetric reproductive interference, the density-dependence factor ( $s$ ) has a similar effect on species coexistence as under asymmetric competition for resources. Conspecific positive density dependence characterized by an intermediate factor $s$ can lead to the loss of the coexistence equilibrium (black zone in Figure 6a–c); in particular, this occurs under asymmetric reproductive interference if there is some symmetric competition for resources (for $α = 0.5$ in Figure 6—figure supplement 2). More importantly, under the two forms of species asymmetry implemented here, conspecific positive density dependence characterized by a high factor $s$ favors the existence of the coexistence equilibrium as a local attractor (Figure 6). Indeed, conspecific positive density dependence characterized by a high factor $s$ reduces the mortality of the least competitive species (as shown in the case of asymmetric competition for resources; see Figure 4d–f) and thereby increases the feasibility of coexistence. This effect is made clear in supplementary analyses where mortality is not density-dependent and where increased $s$ associates with reduced mortality (top rows in Figure 6—figure supplement 3 and Figure 6—figure supplement 4).

In the previous model with asymmetric competition for resources, the coexistence equilibrium is either lost or non-globally stable under conspecific positive density dependence associated with reduced mortality (compared to the coexistence equilibrium when $s$ is close to 0, Figure 4). Here, this form of positive density dependence can substantially increase the coexistence region, compared to the situation without positive density dependence (Figure 6). Note, however, that the condition of global stability is never fulfilled when positive density dependence favors the feasibility of coexistence.

In supplementary analyses, I implemented other values of parameter $α$ (Figure 6—figure supplement 1 and Figure 6—figure supplement 2) or other non-linear or linear mortality functions (Figure 6—figure supplement 3, Figure 6—figure supplement 4; Figure 6—figure supplement 5, and Figure 6—figure supplement 6). Just like in the main analyses, conspecific positive density dependence associated with reduced mortality can substantially increase the coexistence region.

Multispecies models

Models

To finish, I assess the effects of conspecific positive density dependence on the maintenance of species-rich communities in multispecies models. The previous two-species models are combined and modified to account for an arbitrary number of species. The population dynamic of each of the $N$ species is:

\frac{d n_{i}}{d t} = n_{i} [\frac{1}{\sum_{j = 1}^{N} α_{j, i}^{'} \frac{n_{j}}{n_{i}}} - \sum_{j = 1}^{N} α_{j, i} n_{j} - [d + δ_{i} (1 - d)] \times D (n_{i})]

Parameters $α_{j, i}$ and $α_{j, i}^{'}$ represent the strengths of resource competition and reproductive interference experienced by species $i$ because of the presence of species $j$ ( $\forall i$ , $α_{i, i} = 1$ and $α_{i, i}^{'} = 1$ ). Parameter $δ_{i}$ represents the increase in mortality rate of species $i$ compared to the minimum basal mortality rate $d$ . The positive density-dependent mortality function $D (n_{i})$ is modeled as in Equation 1b, and is characterized by its density-dependence factor $s$ .

Four different scenarios are considered successively: (1) only asymmetric competition, (2) only asymmetric reproductive interference, (3) only differences in basal mortality, (4) random communities with all forms of asymmetry. With the first three scenarios, I assess the robustness of the predictions of the two-species models analyzed above, and with the fourth scenario, I test whether conspecific positive density dependence acting on additional mortality can help maintain a high species richness in a random community. In scenarios 2 and 3, I assume that there is symmetric competition for resources ( $α_{i, j} = α$ if $i \neq j$ ), just like in the corresponding two-species models (full methods appear in Supplementary file 1D).

Numerical simulations

For each scenario, 500 species pools of 100 species each were constructed by drawing species parameter values from arbitrary uniform distributions (detailed in Supplementary file 1D; see the robustness to variations in the uniform distributions tested in Figure 7—figure supplement 1). All species were equally abundant initially ( $n_{i} (0) = 1$ ), and simulations were run long enough for initial transients to dissipate. Species were declared extinct if their density fell below $10^{- 3}$ . At the end of each simulation, the number of species remaining was recorded. In all simulations in which multiple species persisted, coexistence occurred at a stable equilibrium.

Consistent with the previous two-species models, the density-dependence factor has a non-linear effect on species richness maintained at equilibrium (Figure 7a–c). Indeed, species richness remains high under the conditions where there is a coexistence equilibrium in the two-species models, including for high density-dependence factors (Figure 4 and Figure 6). Finally, in simulations with random communities of species with all forms of asymmetry, conspecific positive density dependence characterized by a high factor $s$ can favor the existence of stable equilibrium points characterized by the maintenance of many competing species (Figure 7d). Therefore, conspecific positive density dependence acting on extrinsic mortality can help maintain species-rich communities.

Figure 7 with 9 supplements see all

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Species richness maintained in simulated communities.

Each box represents the distribution for 500 simulated communities with 100 species initially. (a) Only asymmetric competition for resources. (b) Only differences in basal mortality. (c) Only asymmetric reproductive interference. (d) Random communities with all forms of species asymmetry. In each subfigure, the range of $s$ is arbitrarily divided in two; one range where there is conspecific positive density dependence at all densities, and another range where positive density-dependent mortality is very low even at very low densities. Minimum basal mortality rate: $d = 0.5$ ; Levels of symmetric competition for resources: $α = 0.05$ (b) or $α = 0$ (c), competition for resources is asymmetric otherwise (**a, d**).

In supplementary analyses, I implemented other values of parameters $d$ and $α$ (Figure 7—figure supplement 2, Figure 7—figure supplement 3, Figure 7—figure supplement 4 and Figure 7—figure supplement 5), other non-linear or linear mortality functions (Figure 7—figure supplement 6 and Figure 7—figure supplement 7), or just a fraction of species undergoing conspecific positive density dependence (Figure 7—figure supplement 8). Just like in the main analyses, conspecific positive density dependence acting on extrinsic mortality can help maintain species-rich communities. This is also the case when species parameter values are drawn from other uniform distributions (Figure 7—figure supplement 1).

A second set of simulations in which species were introduced one at a time produced different results. In such scenarios of community assembly, positive density dependence does not help produce species-rich communities (if species are introduced at very low density; Figure 7—figure supplement 9). This is consistent with the two-species models analyzed where the condition of global stability is not fulfilled for a high density-dependence factor.

Discussion

When intraspecific competition is stronger than interspecific competition, resulting negative density-dependent growth rate has long been recognized as an important factor favoring coexistence among competing species (MacArthur, 1970; Chesson, 2000; McPeek, 2012), and it is therefore not surprising that conspecific positive density dependence acting on intrinsic growth rate can inhibit coexistence among competing species (as shown theoretically by Wang et al., 1999; De Silva and Jang, 2015). Nonetheless, a variety of mechanisms can generate positive density-dependent growth rates (Courchamp et al., 1999; Stephens et al., 1999; Kramer et al., 2009). Contrary to expectations, the models analyzed in this paper suggest that conspecific positive density dependence associated with reduced mortality (e.g. reduced predation via cooperative defense, predator satiation or aposematism; Figure 2) can favor the maintenance of species-rich communities.

How can different forms of conspecific positive density dependence have opposite effects on the feasibility of species coexistence? Conspecific positive density dependence per se promotes species exclusion by reducing the growth rate of inferior competitors that are at lower density than superior competitors. This inhibits species coexistence in previous models accounting for conspecific positive density dependence (Wang et al., 1999; De Silva and Jang, 2015), but also in the models analyzed in this paper (for intermediate factor $s$ ). However, reduced mortality associated with conspecific positive density dependence can also increase the feasibility domain of species coexistence (i.e. it can favor the existence of a locally stable equilibrium of coexistence). This outcome relies on the effect of increased mortality on population dynamics. Increased mortality can lead to the exclusion of less competitive species (Abrams, 1977; Holt, 1985; but see Abrams, 2001), and this is the case in the two models analyzed numerically in this paper (Equations 2 and 3). Therefore, by reducing the mortality of less competitive species, conspecific positive density dependence can favor coexistence among competing species. By contrast, conspecific positive density dependence associated with an increase of mortality should favor species exclusion (e.g. anthropogenic Allee effects; Courchamp et al., 2006). Likewise, in cases where increased mortality increases the feasibility of coexistence (e.g. in a MacArthur’s consumer-resource model; Abrams, 2001), conspecific positive density dependence associated with reduced mortality should also favor species exclusion.

Under conspecific positive density dependence, the least competitive species is often not able to increase in density if it is too rare – i.e. coexistence is only a local attractor and the condition of global stability is not fulfilled. Invasibility is often considered as a fundamental criterion for species coexistence regardless of the underlying mechanism (Chesson, 2000; Siepielski and McPeek, 2010; Grainger et al., 2019); indeed, global stability of a feasible equilibrium point is a sufficient condition for species coexistence. Nonetheless, the condition of global stability is rarely fulfilled in systems with more than two species, and the feasibility of coexistence has also been recognized as an important determinant of multispecies coexistence (see Saavedra et al., 2017; Levine et al., 2017; Grainger et al., 2019, for discussion on the evaluation of multispecies coexistence). While the coexistence equilibria identified do not satisfy this invasibility criterion, conspecific positive density dependence strongly increases the coexistence region (i.e. the feasibility domain of coexistence) in the two-species models analyzed numerically. Moreover, numerical simulations of multispecies models show that conspecific positive density dependence can favor the existence of stable equilibrium points characterized by the maintenance of many competing species. This suggests that positive density dependence acting on mortality can help maintain species diversity in ecological communities.

One aspect that has not been investigated in this study is the implication of stochasticity for the maintenance of species-rich communities (all numerical simulations were deterministic). In the models analyzed in this study, conspecific positive density dependence can only increase the feasibility of a local coexistence equilibrium point that is not robust to stochasticity and strong perturbations. However, another effect of conspecific positive dependence is to increase the species densities at coexistence equilibrium (see Figure 4 and Supplementary file 1A), and therefore to increase robustness to stochastic changes. Under conspecific positive density dependence, only strong stochasticity may lead to local extinction, yet such strong stochasticity would also favor the invasion of new species – the threshold above which a species can invade would be more easily attained by chance. In the face of stochasticity, the effect of conspecific positive density dependence on species richness in metacommunities may therefore differ from the deterministic case studied in this paper, and therefore requires further theoretical investigations (e.g. following the approach followed by Schreiber et al., 2019, combining frequency-dependence and environmental stochasticity).

Putative examples of conspecific positive density dependence acting on extrinsic mortality have already been described in many taxa (Figure 1 and Figure 2) (note however that difficulties in detecting negative density dependence may be similarly applied in detecting positive density-dependence; Detto et al., 2019). This form of conspecific positive density dependence may play an important role in structuring certain ecological communities. For instance, the high species richness that can be found in aposematic organisms at local scale (e.g. in aposematic butterflies, Willmott et al., 2017) is likely to be caused by conspecific positive density dependence associating with reduced mortality (via density-dependent predator avoidance which has received much empirical support; Ruxton et al., 2018). By contrast, an overall pattern of negative density dependence has been observed in other ecological communities (e.g. in many plant communities; Adler et al., 2018). This does not mean that conspecific positive density dependence associated with reduced mortality is not pervasive in those communities. Indeed, it is important to remember that different mechanisms leading to density dependence can occur simultaneously in the same population (Berec et al., 2007). In particular, density-dependent mechanisms can counteract with each other (Feldman and Morris, 2011; Bergamo et al., 2020), and observing an overall pattern of negative density dependence does not mean that some forms of positive density dependence do not take place. In the models analyzed in this paper, conspecific positive density dependence associated with reduced mortality promote species-rich communities in concert with negative density dependence (driven by intraspecific competition for resources), highlighting the necessity of dissecting the different density-dependent mechanisms acting simultaneously (as in Feldman and Morris, 2011; Bergamo et al., 2020). In particular, experimental manipulations of population density coupled with other treatments (e.g. removal of predators/herbivores) could prove useful not only to infer the nature and strength of density dependence but also to define the underlying mechanisms (e.g. this is how negative density dependence in populations of reef fishes has been shown to be driven by competition and predation; Hixon, 1997; Forrester and Steele, 2000; Carr et al., 2002; Holbrook and Schmitt, 2002).

In addition to the assessment of conspecific positive density dependence associated with reduced mortality in nature, the causal link between this form of density dependence and species coexistence remains to be tested empirically. Imposing and amplifying conspecific positive density dependence in a controlled experimental setup could prove fruitful (as in Williams and Levine, 2018, where negative density dependence was manipulated). For instance, within an experimental setup with two competing species, imposing additional mortality (by removing individuals artificially) and enforcing positive density dependence on this source of mortality (with different decline rates of mortality with density; as in Figure 3) could provide information on the implication of conspecific positive density dependence associated with reduced mortality for coexistence. In the theoretical models analyzed here, conspecific positive density dependence strongly promoted the maintenance of species undergoing asymmetric reproductive interference. I thus encourage testing the theoretical predictions presented in this paper on empirical data in organisms involved in such sexual interactions (e.g. Takafuji et al., 1997; Kishi et al., 2009; Takakura et al., 2009; Crowder et al., 2010).

Conclusion

Although a variety of coexistence mechanisms operate simultaneously in nature, these modeling results suggest that conspecific positive density dependence play a considerable role in structuring ecological communities. Conspecific positive density dependence may have opposite effects on species coexistence depending on the underlying density-dependent mechanism. In particular, contrary to expectations, conspecific positive density dependence associated with reduced mortality can help maintain species-rich communities. Hopefully, this theoretical analysis will stimulate further empirical research precisely testing the prevalence of conspecific positive density dependence and investigating its implications for population dynamics, including for coexistence among competing species.

Data availability

All data generated or analysed during this study are included in the manuscript and supporting files. The code (Python) is made available as Source code 1.

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Decision letter

David Donoso

Reviewing Editor; Escuela Politécnica Nacional, Ecuador
Detlef Weigel

Senior Editor; Max Planck Institute for Developmental Biology, Germany

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Acceptance summary:

For decades, negative density dependence (NDD) has been a major paradigm guiding theoretical and experimental research on mechanisms maintaining biodiversity. This paper goes outside the box and provides mathematical support for positive density dependence (PDD) as an additional viable force maintaining biodiversity. By reducing mortality, PDD could limit the exclusion of less competitive species by dominant ones. We are sure that this manuscript will inspire both theoreticians and field biologists aiming to understand how species to coexist in nature.

Decision letter after peer review:

[Editors’ note: the authors submitted for reconsideration following the decision after peer review. What follows is the decision letter after the first round of review.]

Thank you for submitting your work entitled "Positive density-dependence acting on mortality can help maintain species-rich communities" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and a Senior Editor. The reviewers have opted to remain anonymous.

Our decision has been reached after consultation between the reviewers. Based on these discussions and the individual reviews below, we regret to inform you that your work will not be considered further for publication in eLife. We all agree, however, that an improved version of the manuscript (addressing completely reviewer's points below) may be submitted as a new submission in eLife.

All reviewers find merit in your study. And all of them suggest that more theoretical work (like yours) is needed to expand the field, given that positive density-dependence is so often observed in nature. The manuscript is inspiring. For example, one of the reviewers highlighted: "This manuscript is very creative and thought-provoking because it makes me think more about very steep positive density-dependent gradients in a new light, as facilitating coexistence rather than being a unique part of species' ecology". The reviewers also applaud the author's efforts in proposing potentially new mechanisms of species coexistence in light of positive density-dependence. However, the author needs to make more efforts in building biological motivation and checking the robustness of the results. The manuscript is too theoretical/mathematical for the eLife audience and lacks more-biological 'real-life' examples.

Reviewer #1:

I think Aubier accomplished an interesting modeling manuscript on an interesting topic, positive density dependence acting on mortality. However, I feel the manuscript is too theoretical/mathematical for eLife and lacks a more-biological 'real-life' examples.

I do think that a glossary, and a comparative table with NDD on the left, and PDD on the right, then specific patters (like PDD on mortality) on a second row, then possible mechanisms responsible for these patterns, outcomes, and finally examples in real life, could be something to add.

I think the author can do more to have an easily understandable manuscript. All, the mathematical formulas, the graphs, and the text itself are very difficult to follow. Even the code in the zip file is not annotated. This does not have to be this way. Also, some re-ordering may be also appropriate (the first paragraph in the Discussion should likely be the first of the Introduction), and many ideas are repeated two or more times (i.e., Positive density-dependence has been described for most major animal taxa…). Some combinations of words, like existence of coexistence, just make your manuscript unnecessarily difficult to follow. And discussion on mimicry seems just speculation at this point.

Sorry for self-advertizing, but in one of my papers, we forced NDD to Devil's Gardens, one case where ants decrease plant mortality. So patches of monospecific plants accumulate in the forest. We really had a hard time explaining this non-NDD case, (I guess this is then a case of PDD). What I want to say is that there are potentially hundreds of ecologists out there dealing with similar issues in their own system. This manuscript could benefit them. Currently, it does not.

Reviewer #2:

This manuscript is very creative and thought-provoking because it makes me think more about very steep positive density-dependent gradients in a new light, as facilitating coexistence rather than being a unique part of species' ecology. For example, bighorn sheep suffer strong Allee effects because small groups are depredated quickly by mountain lions. This is a terrific example of the Allee effect, but one rarely thinks of this sort of positive density dependence as being a mechanism of species coexistence between large ungulates. This paper makes me think about that.

I do think that the model needs more contextualization in biology from top to bottom. I had to read very carefully, and twice, to understand what I did – for eLife, the paper has to sing so that readers get most of it on the first read. There is an opportunity, and I would argue a necessity, to clarify intra- and interspecific density-dependent processes in the Introduction. The choice of terms in the modeling equations themselves could also be better tied to biology to explain exactly what kinds of systems they represent, and how broadly they can be interpreted. My specific comments are below:

1) Negative density dependence favors coexistence if intraspecific density dependence is stronger than interspecific density dependence. Positive density-dependence seems like a strange mechanism to promote species coexistence when it is interspecific because the most abundant species should only become a stronger and stronger competitor as it rises in abundance. Intraspecific positive density dependence – e.g. Allee effects – seems to make it more difficult for rare species to invade, so it does not promote global stability (subsection “Analytical resolution”, second paragraph). As I understand the analytical part of the model (and I have to be honest, I only read Supplementary file 1A), positive density dependence is only intraspecific because the D(ni) function for each species only includes individuals of that species. s it is always the same between species, though so their density dependence varies perfectly in tandem. I think this is really what struck me as counterintuitive because if I were to think about positive density dependence in the context of competition, I would imagine it to operate interspecifically as well so that the more abundant a species was, the better it would be able to outcompete its competitors (45-47). If I am right about what the model means, then it might be very helpful in the Introduction to explain not just "positive density dependence" writ large, but the different ways it might function (e.g. interspecifically versus intraspecifically, or using specific natural history examples that the model applies to and contrasting them to natural history examples where it does not). The fourth paragraph of the Introduction starts to do this, but are very abstract and don't go into enough detail about differences in kinds of positive density dependence, only its general effects.

2) At very high values of positive frequency dependence, the effects of positive density dependence essentially disappear because the populations hit zero mortality so quickly. I could imagine this being very important in cases where there are for example two aposematic species and both have sufficiently high population sizes that both are protected from predation. Perhaps it would also make sense to imagine two species of pack-hunting carnivores that have zero success alone, but much greater success with even a few individuals, so the two species might be able to coexist as long as both are already present in sufficient abundance. This is sort of an Allee effect too, of course, which are also experienced by species of herd animals that cannot survive in small groups, because they are too vulnerable to predation. But this model seems to describe species that are very similar in their ecology, not just because they are forced to be so to analyze the effects of density dependence, but also because density has identical intraspecific effects. Is this true? If so it would be helpful to explain in the text, and if not, then it would be helpful to explain how broadly it applies.

3) In (2), has the carrying capacity – or effective carrying capacity – also changed? Is the purpose of this model to contrast with (1), where one spp. is a superior competitor, whereas here in (2), one spp. is a superior reproducer/survivor?

4) For Equation 3, I need more hand-holding to explain what's going on. It looks like the new term n2/(n2 + a'1*n1) means the frequency of individuals in the community that is composed of n2 – is that correct? So is this a way of incorporating positive interspecific frequency dependence into your model that already includes positive intraspecific density dependence? Would there be any difference if you modeled reproductive interference as an effect on d rather than on ni, since in a literal way it should affect reproduction rather than carrying capacity? I could also imagine it not matter if its influence is only supposed to be manifest by decreasing dn2/dt, though.

Reviewer #3:

Positive density-dependence is widely observed in nature. It is widely believed that positive density-dependence inhibits species coexistence. The main result of this manuscript is that a positive density-dependence on mortality, under certain forms of population dynamics, increases the feasibility domain of species coexistence. The paper is well-written (but I am not sure if the math here would be easily accessible to the majority of the readership at eLife). I applaud the author's efforts in proposing potentially new mechanisms of species coexistence by positive density-dependence, which could be an important contribution to the coexistence theory. However, the author needs to make more efforts in building biological motivation and checking the robustness of the results.

1) The biological relevance.

– The author motivated the paper by pointing up the ubiquitousness of positive density-dependence (Introduction, fifth paragraph). However, if I understood correctly, most references focus on the Allee effect of the birth rates. Then, is there substantive literature of empirical evidence that positive density-dependence acts on mortality? To be clear, I am not saying that there is not, but to encourage the author to be more explicit about the biological motivations.

– In the fifth paragraph of the Discussion, the author concluded that it is yet impossible to validate the results. I apologize if I misunderstood, but I feel that the paper offers no guidelines for empirical tests. To be clear, I am not asking the author to run analysis on some empirical data, but to encourage the author to discuss how this theory can be potentially tested. Otherwise, it may leave the reader the impression of a math exercise disconnected from ecological nature.

2) The robustness of the results. I understand the following requests can be a bit much, but I sincerely believe that when one proposes a new ecological mechanism, the author needs to prove that it is theoretically robust to convince the empiricists. However, I am happy for a discussion if the author disagrees with anything follows.

– This paper only studies one particular functional form of mortality with positive density-dependence. As the author stated that "assessing the functional form of positive density-dependence in natural populations is tedious", I think the author needs to study other functional forms of positive density-dependence, at least in the 2-species case.

– This paper does not have a sort of null model to estimate the effects of positive density-dependence. That is, whether the effects of positive density-dependence solely comes with the non-linear functional form or because of the positiveness. To do this, I suggest the author add some additional tests on negative density-dependence with the same functional response, to see that the effects are actually coming from positiveness.

– This paper considers the same functional form of positive density-dependence for every species. However, I think it is worth examining some with positive density-dependence and some without in multi-species case.

– To be honest, I am a bit lost why section 1 exists. I thought the author tries to prove that positive density-dependence helps coexistence, but why spent so much time on a particular case where positive density-dependence inhibits coexistence?

– I don't find the parametrization in Supplementary file 1D satisfactory. The range of parameters seems very arbitrary to me. I was puzzled that why no other parametrizations were examined.

https://doi.org/10.7554/eLife.57788.sa1

Author response

[Editors’ note: the authors resubmitted a revised version of the paper for consideration. What follows is the authors’ response to the first round of review.]

All reviewers find merit in your study. And all of them suggest that more theoretical work (like yours) is needed to expand the field, given that positive density-dependence is so often observed in nature. The manuscript is inspiring. For example, one of the reviewers highlighted: "This manuscript is very creative and thought-provoking because it makes me think more about very steep positive density-dependent gradients in a new light, as facilitating coexistence rather than being a unique part of species' ecology". The reviewers also applaud the author's efforts in proposing potentially new mechanisms of species coexistence in light of positive density-dependence. However, the author needs to make more efforts in building biological motivation and checking the robustness of the results. The manuscript is too theoretical/mathematical for the eLife audience and lacks more-biological 'real-life' examples.

I thank the Editors for handling my manuscript and for their positive feedback. I am pleased that all reviewers approved my general aim of investigating the implications of conspecific positive density dependence for species coexistence, and appreciated the counter intuitiveness of my theoretical results. However, it is clear from the feedback I received that I had much more work to do to build biological motivation and to make my approach more impactful. In the revised version of this paper I have sought to address these concerns (i) by thoroughly defining the different types of density dependence (conspecific/heterospecific, negative/positive) that come into play in nature, (ii) by using natural history examples to illustrate the ubiquitousness of conspecific positive density dependence associated with reduced mortality, and (iii) by offering guidelines for testing empirically my theoretical predictions. Collectively, I hope these changes would make my study more accessible to both empiricists and theoreticians.

Additionally, I followed the reviewers’ suggestions of conducting further supplementary analyses (with more than 10 new supplementary figures); altogether, these analyses show that my results are robust to changes in the underlying modeling assumptions.

Reviewer #1:

I think Aubier accomplished an interesting modeling manuscript on an interesting topic, positive density dependence acting on mortality. However, I feel the manuscript is too theoretical/mathematical for eLife and lacks a more-biological 'real-life' examples.

I thank reviewer #1 for this positive feedback. I have followed his/her suggestions, and I have made a considerable effort to make this theoretical study more easily accessible and therefore more impactful. By doing so, I hope that I addressed the reviewer’s concerns successfully.

I do think that a glossary, and a comparative table with NDD on the left, and PDD on the right, then specific patters (like PDD on mortality) on a second row, then possible mechanisms responsible for these patterns, outcomes, and finally examples in real life, could be something to add.

I followed the reviewer’s suggestion to add a Glossary defining terms describing species interactions, density dependence and frequency dependence (negative vs. positive, conspecific vs. heterospecific) (Box 1). In this new glossary but also throughout the manuscript, I am now very clear about the nature of each density-dependent process, and about how species interactions can lead to heterospecific density dependence. I also emphasize that I am investigating cases of conspecific positive density dependence (and not cases of heterospecific positive density dependence caused by mutualistic interspecific interactions) (Box 1).

Following the reviewer’s suggestion, I also added a comparative table (Figure 1). I decided to compare specific patterns of conspecific positive density dependence (associated with increased reproduction or reduced mortality) to emphasize the biological motivation behind my theoretical study on conspecific positive density dependence (note however that negative density dependence is described in Box 1). In this comparative table, I use natural history examples to illustrate each specific patterns (the references can be found in the excellent review papers cited in the caption of Figure 1). Additionally, I added some illustrations of specific natural history examples where conspecific positive density dependence associates with reduced mortality in different taxa (Figure 2) (Introduction).

Altogether, these amendments significantly clarify the manuscript and build the biological motivation behind this theoretical study.

I think the author can do more to have an easily understandable manuscript. All, the mathematical formulas, the graphs, and the text itself are very difficult to follow. Even the code in the zip file is not annotated. This does not have to be this way. Also, some re-ordering may be also appropriate (the first paragraph in the Discussion should likely be the first of the Introduction), and many ideas are repeated two or more times (i.e., Positive density-dependence has been described for most major animal taxa…). Some combinations of words, like existence of coexistence, just make your manuscript unnecessarily difficult to follow. And discussion on mimicry seems just speculation at this point.

In the revised version of the manuscript, I aimed at avoiding useless repetitions and unwieldy combinations of words (e.g., “existence of coexistence” is now referred to as “existence of equilibrium points characterized by species coexistence”). Likewise, I followed the suggestions of moving the paragraph describing the mechanisms favouring species coexistence from the Discussion to the Introduction.

I also removed the discussion on the maintenance of multiple mimicry rings, which was somehow off-topic.

Following the suggestions of the other reviewers, I clarified the presentation of the mathematical formulas by precisely using the terms describing density- or frequency-dependence (also defined in the Glossary; Box 1). In particular, I now precisely explain why I model reproductive interference as a frequency-dependent process acting on the growth rate (subsection “Two-species models with other forms of competitive exclusion”).

I apologize for the non-annotated code that I provided during the initial submission; the code is now fully annotated.

I hope that the modifications I made to address the other reviewers’ concerns also make the manuscript more understandable. Of course, I am open to any other suggestions that would make my study easier to follow.

Sorry for self-advertizing, but in one of my papers, we forced NDD to Devil's Gardens, one case where ants decrease plant mortality. So patches of monospecific plants accumulate in the forest. We really had a hard time explaining this non-NDD case, (I guess this is then a case of PDD). What I want to say is that there are potentially hundreds of ecologists out there dealing with similar issues in their own system. This manuscript could benefit them. Currently, it does not.

In the revised manuscript, I extended the Discussion to fill this gap. First, I advocate the need to assess the occurrence of conspecific positive density dependence in the wild. In particular, if empiricists observe an overall pattern of conspecific negative density dependence (as in many plant communities), some other forms of positive density dependence may still take place at the same time (and may promote species coexistence by their own). Second, I offer guidelines for testing empirically my theoretical predictions on the link between conspecific positive density dependence and coexistence. Altogether, I hope these amendments will be helpful to empiricists interested in the theoretical prediction presented in my paper.

Reviewer #2:

This manuscript is very creative and thought-provoking because it makes me think more about very steep positive density-dependent gradients in a new light, as facilitating coexistence rather than being a unique part of species' ecology. For example, bighorn sheep suffer strong Allee effects because small groups are depredated quickly by mountain lions. This is a terrific example of the Allee effect, but one rarely thinks of this sort of positive density dependence as being a mechanism of species coexistence between large ungulates. This paper makes me think about that.

I do think that the model needs more contextualization in biology from top to bottom. I had to read very carefully, and twice, to understand what I did – for eLife, the paper has to sing so that readers get most of it on the first read. There is an opportunity, and I would argue a necessity, to clarify intra- and interspecific density-dependent processes in the Introduction. The choice of terms in the modeling equations themselves could also be better tied to biology to explain exactly what kinds of systems they represent, and how broadly they can be interpreted. My specific comments are below:

I thank reviewer #2 for this positive feedback; I am glad that my study made him/her think about density dependence and coexistence in a new light. In the revised manuscript, I aimed at making this theoretical study more easily accessible by using some “real-life” examples. I also followed the reviewer #1’s suggestion to add a Glossary defining terms describing species interactions, density dependence and frequency dependence. I carefully used those terms to describe the modeling equations, making them better tied to the biological processes. I hope that by doing so, I addressed the reviewer #2’s concerns successfully.

1) Negative density dependence favors coexistence if intraspecific density dependence is stronger than interspecific density dependence. Positive density-dependence seems like a strange mechanism to promote species coexistence when it is interspecific because the most abundant species should only become a stronger and stronger competitor as it rises in abundance. Intraspecific positive density dependence – e.g. Allee effects – seems to make it more difficult for rare species to invade, so it does not promote global stability (subsection “Analytical resolution”, second paragraph). As I understand the analytical part of the model (and I have to be honest, I only read Supplementary file 1A), positive density dependence is only intraspecific because the D(ni) function for each species only includes individuals of that species. s it is always the same between species, though so their density dependence varies perfectly in tandem. I think this is really what struck me as counterintuitive because if I were to think about positive density dependence in the context of competition, I would imagine it to operate interspecifically as well so that the more abundant a species was, the better it would be able to outcompete its competitors (45-47). If I am right about what the model means, then it might be very helpful in the Introduction to explain not just "positive density dependence" writ large, but the different ways it might function (e.g. interspecifically versus intraspecifically, or using specific natural history examples that the model applies to and contrasting them to natural history examples where it does not). The fourth paragraph of the Introduction starts to do this, but are very abstract and don't go into enough detail about differences in kinds of positive density dependence, only its general effects.

By using the terms defined in the Glossary (Box 1), I clarified the nature of each density-dependent process, including conspecific positive density dependence. In particular, as correctly explained by the reviewer #2, I modeled conspecific positive density dependence by including only the density of the focal species in the term D(ni). Nonetheless, this does not mean that species densities “varies perfectly in tandem” through this mortality term. Indeed, if one species is more abundant than the other, then mortality rates are different (because species densities are different). Conspecific positive density dependence is not interspecific but it can affect species coexistence by changing the densities determining the strengths of competition or reproductive interference. I hope this is now clarified.

Following the suggestions of reviewer #1, I also tried to clarify the nature of conspecific positive density dependence by using a comparative table (Figure 1) and specific natural history examples (Figure 2) (Introduction).

2) At very high values of positive frequency dependence, the effects of positive density dependence essentially disappear because the populations hit zero mortality so quickly. I could imagine this being very important in cases where there are for example two aposematic species and both have sufficiently high population sizes that both are protected from predation. Perhaps it would also make sense to imagine two species of pack-hunting carnivores that have zero success alone, but much greater success with even a few individuals, so the two species might be able to coexist as long as both are already present in sufficient abundance. This is sort of an Allee effect too, of course, which are also experienced by species of herd animals that cannot survive in small groups, because they are too vulnerable to predation. But this model seems to describe species that are very similar in their ecology, not just because they are forced to be so to analyze the effects of density dependence, but also because density has identical intraspecific effects. Is this true? If so it would be helpful to explain in the text, and if not, then it would be helpful to explain how broadly it applies.

I thank reviewer #2 for this succinct and accurate summary of the reason why conspecific positive density dependence associated with reduced mortality can favor coexistence. Reviewer #2 is also right about the modeling assumption; species have identical intraspecific effects (this is now clarified in manuscript; subsection “Two-species model with asymmetric competition for resources”). But one must remember that even though the functions describing intraspecific effects are identical in the different species, if species differ in their density (e.g. via interspecific effects) then their changes in density due to intraspecific effects differ. Therefore, the model exactly consider the situation described by the reviewer at the beginning of this comment.

Following one of the reviewer #3’s suggestions, however, I relaxed this assumption by conducting simulations where only a fraction of species incur conspecific positive density dependence, and I showed that the outcome is qualitatively similar (Figure 7—figure supplement 8; subsection “Numerical simulations”).

3) In (2), has the carrying capacity – or effective carrying capacity – also changed? Is the purpose of this model to contrast with (1), where one spp. is a superior competitor, whereas here in (2), one spp. is a superior reproducer/survivor?

Exactly, one species is a superior survivor in model (2) and one species is a superior reproducer in model (3); and the effective carrying capacity – density values reached at equilibrium without competitor – remains the same for the two species (just like all other assumptions made in the first model). I clarified this last point (subsection “Two-species models with other forms of competitive exclusion”).

4) For Equation 3, I need more hand-holding to explain what's going on. It looks like the new term n2/(n2 + a'1*n1) means the frequency of individuals in the community that is composed of n2 – is that correct? So is this a way of incorporating positive interspecific frequency dependence into your model that already includes positive intraspecific density dependence? Would there be any difference if you modeled reproductive interference as an effect on d rather than on ni, since in a literal way it should affect reproduction rather than carrying capacity? I could also imagine it not matter if its influence is only supposed to be manifest by decreasing dn2/dt, though.

In the revised manuscript, the term of reproductive interference is now transformed in a way that make it clearer (though this is mathematically the exact same term) because it make it explicit that the frequency of heterospecifics (n1/n2) reduces the intrinsic growth rate (heterospecific negative frequency dependence, Box 1). I also present this mathematical term more thoroughly in the text (subsection “Two-species models with other forms of competitive exclusion”).

Given that reproductive interference reduces the intrinsic growth rate (e.g., the first term within the brackets = 1 in species 1 is not affected by reproductive interference; this is now clearer in the revised form of the equation), we cannot model this process as a change in the term of mortality rate.

Reviewer #3:

Positive density-dependence is widely observed in nature. It is widely believed that positive density-dependence inhibits species coexistence. The main result of this manuscript is that a positive density-dependence on mortality, under certain forms of population dynamics, increases the feasibility domain of species coexistence. The paper is well-written (but I am not sure if the math here would be easily accessible to the majority of the readership at eLife). I applaud the author's efforts in proposing potentially new mechanisms of species coexistence by positive density-dependence, which could be an important contribution to the coexistence theory. However, the author needs to make more efforts in building biological motivation and checking the robustness of the results.

I thank reviewer #3 for this succinct (and accurate) summary and for this positive feedback. In the revised manuscript, I made considerate effort in building biological motivation (by using natural history examples), in checking the robustness of the results (with more than 10 new supplementary figures), and in offering guidelines for testing empirically my theoretical predictions. I hope that I addressed the reviewer’s concerns successfully.

1) The biological relevance.

– The author motivated the paper by pointing up the ubiquitousness of positive density-dependence (Introduction, fifth paragraph). However, if I understood correctly, most references focus on the Allee effect of the birth rates. Then, is there substantive literature of empirical evidence that positive density-dependence acts on mortality? To be clear, I am not saying that there is not, but to encourage the author to be more explicit about the biological motivations.

Following one of reviewer #1’s suggestions, I added a comparative table to describe the different types of conspecific positive density dependence (Figure 1), and I included some natural history examples (Figure 2). While there are many putative cases of such density dependence, whether this form of density dependence is pervasive in ecological communities remains an open question. This is now discussed in the Discussion. In particular, I emphasize that this form of positive density dependence may take place even if we observe an overall pattern of conspecific negative density dependence (as in many plant communities). Even if this is not the dominant density dependent force, it can still promote species coexistence as predicted by the theoretical models.

– In the fifth paragraph of the Discussion, the author concluded that it is yet impossible to validate the results. I apologize if I misunderstood, but I feel that the paper offers no guidelines for empirical tests. To be clear, I am not asking the author to run analysis on some empirical data, but to encourage the author to discuss how this theory can be potentially tested. Otherwise, it may leave the reader the impression of a math exercise disconnected from ecological nature.

Following the reviewer’s suggestion, I now offer guidelines for assessing the prevalence of conspecific positive density dependence and for testing empirically its implication for coexistence (Discussion). Given that I got the most striking results in the case with asymmetric reproductive interference, I encourage empiricists working on organisms undergoing reproductive interference to test the implication of conspecific positive density dependence on coexistence.

2) The robustness of the results. I understand the following requests can be a bit much, but I sincerely believe that when one proposes a new ecological mechanism, the author needs to prove that it is theoretically robust to convince the empiricists. However, I am happy for a discussion if the author disagrees with anything follows.

By running new simulations based on the reviewer’s suggestions, I confirmed that my results are robust to changes in modelling assumption. I detail each of those supplementary analyses below.

– This paper only studies one particular functional form of mortality with positive density-dependence. As the author stated that "assessing the functional form of positive density-dependence in natural populations is tedious", I think the author needs to study other functional forms of positive density-dependence, at least in the 2-species case.

This is a very good point. To check the robustness of my predictions to the functional form of density dependence, I conducted simulations with other linear or non-linear mortality function (represented in Figure 3—figure supplement 1; note that I could not test γ values >3; because the numerical method was not reliable in that condition). I got similar results as in the main analyses in the two-species models (subsection “Analytical resolution”; subsection “Numerical resolutions; Figure 4—figure supplements 3 and 4, Figure 6—figure supplements 3-6) but also in the multispecies models (subsection “Numerical simulations”; Figure 7—figure supplements 6 and 7).

– This paper does not have a sort of null model to estimate the effects of positive density-dependence. That is, whether the effects of positive density-dependence solely comes with the nonlinear functional form or because of the positiveness. To do this, I suggest the author add some additional tests on negative density-dependence with the same functional response, to see that the effects are actually coming from positiveness.

I acknowledge that there was no sort of null model. In the revised version, however, I now run simulations where parameter s associates with reduced mortality and not density dependence. The predictions are similar as in the main analysis, highlighting that reduced mortality leads promotes species coexistence (subsection “Numerical resolutions”; Figure 6—figure supplements 3-4), as emphasized in the Abstract and the Introduction.

– This paper considers the same functional form of positive density-dependence for every species. However, I think it is worth examining some with positive density-dependence and some without in multi-species case.

I conducted simulations with the multispecies model (accounting for random communities) where only a fraction of species undergo conspecific density dependence. I get qualitatively the same result (subsection “Numerical simulations”; Figure 7—figure supplement 8).

– To be honest, I am a bit lost why section 1 exists. I thought the author tries to prove that positive density-dependence helps coexistence, but why spent so much time on a particular case where positive density-dependence inhibits coexistence?

I still believe that this is necessary to start the study by the analysis of this model. The first model based on Lotka equations in their purest form is analyzed analytically, precisely pinpointing the effect of the positive density-dependence factor on coexistence. While positive density-dependence does not promote coexistence in this case, we observe a recover of the coexistence equilibrium via reduced mortality, and this is exactly how conspecific positive density dependence promotes coexistence in the other models. Therefore, I think that the time spent in reading this particular case then eases the reading of the two other parts (precisely because the underlying mechanism promoting the recover of the coexistence equilibrium is made clear). This is why I have not changed the structure of the paper in the revised manuscript; I hope the reviewer appreciates my point.

– I don't find the parametrization in Supplementary file 1D satisfactory. The range of parameters seems very arbitrary to me. I was puzzled that why no other parametrizations were examined.

I conducted simulations with other range of parameters. I get qualitatively the same result(subsection “Numerical simulations”; Figure 7—figure supplement 1).

https://doi.org/10.7554/eLife.57788.sa2

Article and author information

Author details

Thomas G Aubier

Department of Evolutionary Biology and Environmental Studies, University of Zurich, Zurich, Switzerland

Contribution
Conceptualization, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing

For correspondence
thomas.aubier@normalesup.org

Competing interests
No competing interests declared

"This ORCID iD identifies the author of this article:" 0000-0001-8543-5596

Funding

No external funding was received for this work.

Acknowledgements

I am very grateful to M Dubard, T Koffel and P Quévreux for comments on early versions of the manuscript. I also thank David Donoso and three anonymous reviewers for comments that have helped improve this manuscript.

Senior Editor

Detlef Weigel, Max Planck Institute for Developmental Biology, Germany

Reviewing Editor

David Donoso, Escuela Politécnica Nacional, Ecuador

Version history

Received: April 22, 2020
Accepted: May 22, 2020
Version of Record published: June 18, 2020 (version 1)

Copyright

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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Thomas G Aubier

(2020)

Positive density dependence acting on mortality can help maintain species-rich communities

eLife 9:e57788.

https://doi.org/10.7554/eLife.57788

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Box 1.

Mechanisms causing conspecific positive density dependence, and affecting reproduction or mortality.

Example of organisms undergoing conspecific positive density dependence that associates with reduced mortality.

Non-linear mortality functions with different density-dependence factors (s).

Effects of asymmetric competition for resources (α1) and positive density-dependence factor (s) on coexistence.

Effect of the positive density-dependence factor (s) on the zero-net-growth isoclines in the case of asymmetric competition for resources.

Effects of difference in basal mortality between species (δ) (a,b,c) or asymmetric reproductive interference (α1′) (d,e,f), and positive density-dependence factor (s) on coexistence, using the numerical resolution method (systems of Equations 2 and 3, respectively).

Species richness maintained in simulated communities.

Author details

Thomas G Aubier

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For correspondence

Competing interests

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Further reading

Non-linear mortality functions with different density-dependence factors ( $s$ ).

Effects of asymmetric competition for resources ( $α_{1}$ ) and positive density-dependence factor ( $s$ ) on coexistence.

Effect of the positive density-dependence factor ( $s$ ) on the zero-net-growth isoclines in the case of asymmetric competition for resources.

Effects of difference in basal mortality between species ( $δ$ ) (a,b,c) or asymmetric reproductive interference ( $α_{1}^{'}$ ) (d,e,f), and positive density-dependence factor ( $s$ ) on coexistence, using the numerical resolution method (systems of Equations 2 and 3, respectively).