SIAM News Blog

Maintaining Biodiversity in an Increasingly Variable World

By Sebastian J. Schreiber

All living organisms experience fluctuations in environmental conditions like temperature, precipitation, nutrient availability, and predation risk. Because these conditions impact survival, growth, and reproduction, the environmental fluctuations cause additional fluctuations in population densities. As a result, environmental fluctuations can affect the viability of populations and the dynamics of interacting species. The most recent report from the Intergovernmental Panel on Climate Change found that the frequency and intensity of heavy precipitation events—as well as concurrent heatwaves and droughts—have increased since the 1950s. Understanding the ecological impacts of these growing environmental variabilities is critical for managing populations and conserving biodiversity. Recent mathematical developments in the analysis of stochastic models [2, 4, 5] allow researchers to study these impacts.

Environmental fluctuations’ effect on populations is known as environmental stochasticity. “[It is] obviously true that the numbers in most natural populations are sometimes increasing and sometimes decreasing, and that these fluctuations may be enormous,” Herbert Andrewartha wrote in the 1950s. “One is thus led to expect that the so-called stochastic models of populations might be more realistic and therefore more successful than the deterministic models” [1]. Richard Lewontin and Dan Cohen answered Andrewartha’s call [7] and studied the simplest possible model that accounts for environmental stochasticity:


Here, \(x(t)\) is the population density at time \(t\) and \(f(t)\) corresponds to the fitness of individuals at time \(t\) (the average contribution of an individual to the population size over a single time step). If \(f(t)\) is a sequence of independent and identically distributed (i.i.d.) random variables, then \(\text{lim}_{t\rightarrow \infty}\frac{1}{t}\ln x(t)=\mathbb{E}[\ln f(1)]\) with probability \(1\) by the law of large numbers. The population thus grows exponentially quickly if the per capita growth rate \(r=\mathbb{E}[\ln r]\) is positive; if \(r\) is negative, the population decays exponentially quickly toward extinction. In contrast, the expected population size \(\mathbb{E}[n(t)]\) grows exponentially like \(\mathbb{E}[f(1)]^t\). Lewontin and Cohen highlighted a surprising prediction from this simple model, based on the fact that the geometric mean \(\exp(\mathbb{E}[\ln f(1)])\) is less than the arithmetic mean \(\mathbb{E}[f(1)]\): They noted that “even though the expectation of population size may grow infinitely large with time, the probability of extinction may approach unity” [7].

Mathematical models of population growth often account for density dependence in the fitness \(f\) of the population:


where \(\xi(t)\) is a sequence of i.i.d. random variables representing demographic impacts of environmental fluctuations. Under appropriate assumptions about the behavior near infinity, these models exhibit statistically bounded fluctuations. Furthermore, if the per capita growth rate when rare (GRWR) \(r=\mathbb{E}[\ln f(0, \xi(1))]\) is positive, then the population is stochastically persistent — it tends to spend little time near \(0\). Alternatively, the population tends toward extinction if \(r<0\). This simple characterization of extinction versus persistence provides a mathematic proof that increased variability in precipitation may have led to the local extinction of two populations of Bay checkerspot butterflies [8] (see Figure 1).

Figure 1. Increased environmental stochasticity promotes extinction. Simulations of a data-based model of form (2) for Bay checkerspot butterflies that are experiencing variability in precipitation. 1a. For precipitation variability before 1971, \(r>0\) and stochastic persistence occurs. 1b. For precipitation variability after 1971, \(r<0\) and asymptotic extinction occurs. These predictions are consistent with the hypothesis that increased precipitation variability caused the local extinction of two butterfly populations [8]. Figure courtesy of Sebastian Schreiber.

Species are not isolated from each other; they interact through a complex web of direct and indirect pathways. Moreover, individuals within a species may differ from one another in demographically important ways due to variations in behavior, morphology, physiology, or spatial location. To account for diversity of species interaction and individual differences, consider Markovian models in which \(x=(x_1,...,x_n)\) is the vector of species’ densities and \(y \in \mathbb{R}^k\) are auxiliary variables [4]:

\[x_i(t+1)=x_i(t)f_i(x(t),y(t),\xi(t+1))\] \[i=1,...,n\tag3\] \[y(t+1)=G(x(t),y(t),\xi(t+1)).\]

The auxiliary variables describe population structure (i.e., keep track of each species’ frequency in a patch, age class, or stage), capture feedback variables (e.g., trait evolution and plant-soil feedbacks), or allow for structure in environmental fluctuations (e.g., autocorrelation).

Characterizing coexistence and extinction in these types of models is a much more delicate process than in the single-species model. However, studies have extended an approach that is inspired by Josef Hofbauer’s work with deterministic models [6] to these stochastic models, as well as to stochastic differential equations and piecewise deterministic Markov processes (PDMPs) [2, 4, 5]. Like the single-species models, this approach relies on GRWRs. But unlike the single-species models, there are multiple contexts in which a species may become rare. These contexts are given by ergodic measures \(\mu(dxdy)\) of \((3)\) that support a subset of species, i.e., \(\mu(\{(x,y):\min_ix_i=0\})=1\). For such an ergodic measure, the GRWR of a missing species \(i\) when introduced at infinitesimally small densities is

\[r_i(\mu)=\int\mathbb{E}[\ln f_i(x,y, \xi(1))]\mu(dxdy).\]

This is the per capita growth rate \(\ln f_i\) averaged over the fluctuations in \(x\), \(y\), and \(\xi\). The Hofbauer condition ensures coexistence (i.e., all species’ densities tend to stay away from low values) if fixed weights \(w_i>0\) exist, such that

\[\sum_i w_i r_i(\mu)>0 \enspace \text{for} \enspace \textrm{all} \enspace \textrm{ergodic} \enspace \mu \enspace \textrm{with} \enspace \mu({(x,y): \min_i x_i=0})=1.\tag4\]

The function \(V(x,y)=-\Sigma_i w_i \ln x_i\) then acts like a type of average Lyapunov function near the extinction set — i.e., \(V\) tends to increase along trajectories when \(\min_i x_i\) is sufficiently small. Related conditions help researchers identify when one or more species goes towards extinction exponentially quickly. These results collectively allow one to determine whether all modeled species coexist, or whether one or multiple stochastic attractors are present wherein one or more species are excluded.

Figure 2. Autocorrelated environmental fluctuations alter ecological outcomes. 2a. In a deterministic model of two competing species, one species always excludes the other. 2b-2d. Autocorrelated fluctuations in demographic parameters can alter this outcome, allowing for stochastic coexistence (2b) or a stochastic bistability (2c and 2d) for which there is a positive probability of losing either species for the same initial conditions [9]. Figure courtesy of Sebastian Schreiber.

Application of these results to multispecies models has yielded new, mathematically rigorous insights into environmental stochasticity’s impact on the dynamics of communities of interacting species. For example, one study examines the way in which autocorrelated fluctuations in fecundity or survival determine the fate of competing species [9]. For these models, \(f_i(x,y)=\lambda_i(y_i)/(1+x_i+x_2)+s_i(y_i)\), where \(\lambda_i\) and \(s_i\) are the maximal fecundity and survivorship of species \(i\) and \(y=(y_1,y_2)\) is a multivariate autoregressive process. In the absence of environmental fluctuations (i.e., when \(y\) remains constant), the species with the higher reproductive number \(\lambda_i/(1-s_i)\) excludes the other (see Figure 2a). Accounting for environmental fluctuations can shift the exclusionary dynamics to stochastic coexistence (see Figure 2b) or a stochastic priority effect (see Figure 2c and 2d), whereby there is a positive probability that one species drives the other to extinction. These outcomes have a delicate dependence on whether survival or fecundity fluctuates, and the sign of its temporal autocorrelation. For example, positively autocorrelated fluctuations in fecundity promote coexistence and positively autocorrelated fluctuations in survival promote stochastic bistabilities. Equally surprising is Michel Benaïm and Claude Lobry’s use of PDMP Lotka-Volterra models to show that random switching between two environments—both of which favor the species in question—can lead to that species’ extinction [3].

Indirect species interactions are more challenging to study but can yield additional unexpected results. For example, analyses of the dynamics of two prey species that share a common predator revealed that an increase in the density of one prey species leads to an increase in the predator’s density and a corresponding decrease in the other prey species [10, 11]. Although the prey are not actually competing, they appear to be due to the indirect effect of the predator. A simple rule of dominance emerges in the absence of autocorrelated environmental fluctuations: the prey species that supports the higher mean predator density excludes the other prey [11]. Autocorrelated fluctuations in predator attack rates can shift the exclusionary dynamics to stochastic coexistence or a stochastic priority effect [10]. Yet unlike the competition models, the intermediary species (the predator) may exhibit varied responses to environmental perturbations on shorter and longer time scales. Highly autocorrelated environmental fluctuations can therefore generate different ecological outcomes (stochastic bistability) than those that result from weakly autocorrelated environmental fluctuations (coexistence).

Despite this progress, many challenges remain. Significant gaps endure between the necessary and sufficient conditions for stochastic coexistence, and a deeper biological understanding of environmental stochasticity’s impact on the ecological dynamics of more diverse communities remains elusive. However, a growing number of talented mathematicians are currently tackling these problems; further insight is sure to follow.

[1] Andrewartha, H.G. (1957). The use of conceptual models in population ecology. Cold Spring Harbor Symp. Quant. Biol., 22, 219-236.
[2] Benaïm, M. (2018). Stochastic persistence. Preprint, arXiv:1806.08450.
[3] Benaïm, M., & Lobry, C. (2016). Lotka-Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder.” Annals Appl. Prob., 26, 3754-3785.
[4] Benaïm, M., & Schreiber, S.J. (2019). Persistence and extinction for stochastic ecological models with internal and external variables. J. Math. Biol., 79, 393-431.
[5] Hening, A., & Nguyen, D.H. (2018). Coexistence and extinction for stochastic Kolmogorov systems. Annals Appl. Prob., 28, 1893-1942.
[6] Hofbauer, J. (1981). A general cooperation theorem for hypercycles. Monatshefte für Mathematik, 91, 233-240.
[7] Lewontin, R.C., & Cohen, D. (1969). On population growth in a randomly varying environment. Proc. Nat. Acad. Sci., 62, 1056-1060.
[8] McLaughlin, J.F., Hellmann, J.J., Boggs, C.L., & Ehrlich, P.R. (2002). Climate change hastens population extinctions. Proc. Nat. Acad. Sci., 99, 6070-6074.
[9] Schreiber, S.J. (2021). Positively and negatively autocorrelated environmental fluctuations have opposing effects on species coexistence. Amer. Naturalist, 197(4), 405-414.
[10] Schreiber, S.J. (2021). Temporally auto-correlated predator attacks structure ecological communities. Preprint, bioRxiv.
[11] Schreiber, S.J. (2021). The P* rule in the stochastic Holt-Lawton model of apparent competition. Discrete Contin. Dynam. Syst. B, 26(1), 633-644.

Sebastian J. Schreiber is a professor of evolution and ecology at the University of California, Davis. He uses the theories of stochastic processes and dynamical systems to tackle questions in ecology, evolution, and epidemiology.

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