Coexistence in a fluctuating environment by the effect of relative nonlinearity: A minimal model

Abstract

The minimal model of the "relative nonlinearity" type fluctuation-maintained coexistence is investigated. The competing populations are affected by an environmental white noise. With quadratic density dependence, the long-term growth rates of the populations are determined by the average and the variance of the (fluctuating) total density. At most two species can coexist on these two "regulating" variables; competitive exclusion would ensue in a constant environment. A numerical study of the expected time until extinction of any of the two species reveals that the criterion of mutual invasibility predicts the parameter-range of long-term coexistence correctly in the limit of zero extinction threshold. However, any extinction threshold consistent with a realistic population size will allow only short-term coexistence. Therefore, our simulations question the biological relevance of mutual invasibility, as a sufficient condition of coexistence, for large density fluctuations. We calculate the average and the variance of the fluctuating density of the coexisting populations analytically via the moment-closure approximation; the results are reasonably close to the simulated behavior. Based on this treatment, robustness of coexistence is studied in the limit of infinite population size. We interpret the results of this analysis in the context of necessity of niche segregation with respect to the regulating variables using a framework theory published earlier.

Chesson's coexistence theory

“…The time averages then read and , where V is the variance of the resource. Quite literally, species 1 consumes the mean and species 2 the variance of the resource (Levins , Kisdi and Meszéna , Szilágyi and Meszéna ); formally, and F 2 = V act as two separate limiting factors. As such, we have a choice to make: which one should we treat as the baseline?…”

Sensitivity analysis of coexistence in ecological communities: theory and application

“…However, the same methodology may be extended to more complex dynamical states, like limit cycles (Barabás et al . ; Barabás & Ostling ) or aperiodic stationary oscillations (Szilágyi & Meszéna ), both in discrete and continuous time. One may also consider communities where the species have complex life cycles, requiring structured population models (Szilágyi & Meszéna ; Barabás et al .…”

“…() presented a new approach for studying the robustness of coexistence and offered a theoretical framework for the construction of community‐wide sensitivity formulae which explicitly quantify the response of population abundances to perturbations of arbitrary model parameters. Recently, a series of such formulae have been worked out for non‐equilibrium communities and communities of structured populations within this framework ( Szilágyi & Meszéna , b, ; Barabás et al . , b, ; Barabás & Ostling ; Barabás et al .…”

A general condition for adaptive genetic polymorphism in temporally and spatially heterogeneous environments

“…It was therefore concluded that temporal fluctuations can only account for a limited amount of the observed genetic variance in diploid populations (Dempster 1955) and that there is no tendency to maintain polymorphism in haploid populations (Cook and Hartl 1974). However, two mechanisms were subsequently identified that can maintain genetic polymorphism under temporal fluctuations, named the "storage effect of generation overlap" (Chesson and Warner 1981;Chesson 1984) and the "effect of relative non-linearity" (Chesson 1994;Szilagyi and Meszena 2010). In both cases, selection is no longer a function of time alone: With the storage effect, selection acts only on a short-lived stage of the life cycle (e.g., juveniles), while a long-lived stage (e.g., adults or persistent dormant stages) is not affected by the fluctuations.…”