Persistence in Stochastic Lotka–Volterra Food Chains with Intraspecific Competition

Identifying and explaining resilience in ecological networks

Resilient ecological systems are more likely to persist and function in the Anthropocene. Current methods for estimating an ecosystem's resilience rely on accurately parameterized ecosystem models, which is a significant empirical challenge. In this paper, we adapt tools from biochemical kinetics to identify ecological networks that exhibit 'structural resilience', a strong form of resilience that is solely a property of the network structure and is independent of model parameters. We undertake an exhaustive search for structural resilience across all three-species ecological networks, under a generalized Lotka-Volterra modelling framework. Out of 20,000 possible network structures, approximately 2% display structural resilience. The properties of these networks provide important insights into the mechanisms that could promote resilience in ecosystems, provide new theoretical avenues for qualitative modelling approaches and provide a foundation for identifying robust forms of ecological resilience in large, realistic ecological networks.

Coevolution of Patch Selection in Stochastic Environments

AbstractSpecies interact in landscapes where environmental conditions vary in time and space. This variability impacts how species select habitat patches. Under equilibrium conditions, evolution of this patch selection can result in ideal free distributions where per capita growth rates are zero in occupied patches and negative in unoccupied patches. These ideal free distributions, however, do not explain why species occupy sink patches, why competitors have overlapping spatial ranges, or why predators avoid highly productive patches. To understand these patterns, we solve for coevolutionarily stable strategies (coESSs) of patch selection for multispecies stochastic Lotka-Volterra models accounting for spatial and temporal heterogeneity. In occupied patches at the coESS, we show that the differences between the local contributions to the mean and the variance of the long-term population growth rate are equalized. Applying this characterization to models of antagonistic interactions reveals that environmental stochasticity can partially exorcize the ghost of competition past, select for new forms of enemy-free and victimless space, and generate hydra effects over evolutionary timescales. Viewing our results through the economic lens of modern portfolio theory highlights why the coESS for patch selection is often a bet-hedging strategy coupling stochastic sink populations. Our results highlight how environmental stochasticity can reverse or amplify evolutionary outcomes as a result of species interactions or spatial heterogeneity.

Strong stochastic persistence of some Lévy-driven Lotka-Volterra systems

We study a class of Lotka-Volterra stochastic differential equations with continuous and pure-jump noise components, and derive conditions that guarantee the strong stochastic persistence (SSP) of the populations engaged in the ecological dynamics. More specifically, we prove that, under certain technical assumptions on the jump sizes and rates, there is convergence of the laws of the stochastic process to a unique stationary distribution supported far away from extinction. We show how the techniques and conditions used in proving SSP for general Kolmogorov systems driven solely by Brownian motion must be adapted and tailored in order to account for the jumps of the driving noise. We provide examples of applications to the case where the underlying food-web is: (a) a 1-predator, 2-prey food-web, and (b) a multi-layer food-chain.

Interpretation and Dynamics of the Lotka–Volterra Model in the Description of a Three-Level Laser

In this work, the Lotka–Volterra equations where applied to laser physics to describe population inversion and the number of emitted photons. Given that predation and stimulated emissions are analogous processes, two rate equations where obtained by finding suitable parameter transformations for a three-level laser. This resulted in a set of differential equations which are isomorphic to several laser models under accurate parameter identification. Furthermore, the steady state provided two critical points: one where light amplification stops and another where continuous-wave operation is achieved. Lyapunov’s first method of stability yielded the conditions for the convergence to the continuous-wave point, whereas a Lyapunov potential provided its stability regions. Finally, the Q-Switching technique was modeled by introducing a periodic variation of the quality Q of the cavity. This resulted in the transformation of the asymptotically stable fixed point into a limit cycle in the phase space.

Cooperative Equilibrium in Biosphere Evolution: Reconciling Competition and Cooperation in Evolutionary Ecology

As our understanding of biological evolution continues to deepen, tension still surrounds the relationship between competition and cooperation in the evolution of the biosphere, with rival viewpoints often associated with the Red Queen and Black Queen hypotheses respectively. This essay seeks to reconcile these viewpoints by integrating observations of some general trends in biosphere evolution with concepts from game theory. It is here argued that biodiversity and ecological cooperation are intimately related, and that both tend to cyclically increase over biological history; this is likely due to the greater relative stability of cooperation over competition as a means of long-term conflict resolution within ecosystems. By integrating this view of the biosphere with existing models such as Niche Game Theory, it may be argued that competition and cooperation in ecosystems coexist at equilibria which shift preferentially towards increasing cooperation over biological history. This potentially points to a state of "cooperative equilibrium" as a limit or endpoint in long-term biosphere evolution, such that Black Queen and Red Queen behavior dominate different phases in an evolutionary movement towards optimal cooperative stability in ecological networks. This concept, if accepted, may also bear implications for developing future mathematical models in evolutionary biology, as well as for resolving the perennial debate regarding the relative roles of conflict and harmony in nature.

Stationary distributions of persistent ecological systems

We analyze ecological systems that are influenced by random environmental fluctuations. We first provide general conditions which ensure that the species coexist and the system converges to a unique invariant probability measure (stationary distribution). Since it is usually impossible to characterize this invariant probability measure analytically, we develop a powerful method for numerically approximating invariant probability measures. This allows us to shed light upon how the various parameters of the ecosystem impact the stationary distribution. We analyze different types of environmental fluctuations. At first we study ecosystems modeled by stochastic differential equations. In the second setting we look at piecewise deterministic Markov processes. These are processes where one follows a system of differential equations for a random time, after which the environmental state changes, and one follows a different set of differential equations-this procedure then gets repeated indefinitely. Finally, we look at stochastic differential equations with switching, which take into account both the white noise fluctuations and the random environmental switches. As applications of our theoretical and numerical analysis, we look at competitive Lotka-Volterra, Beddington-DeAngelis predator-prey, and rock-paper-scissors dynamics. We highlight new biological insights by analyzing the stationary distributions of the ecosystems and by seeing how various types of environmental fluctuations influence the long term fate of populations.

A general theory of coexistence and extinction for stochastic ecological communities

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

The competitive exclusion principle in stochastic environments

In its simplest form, the competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. However, it has been observed empirically that in some settings it is possible to have coexistence. One example is Hutchinson's 'paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times. In this paper we show in various settings how a variable (stochastic) environment enables a set of competing species limited by a smaller number of resources or other density dependent factors to coexist. If the environmental fluctuations are modeled by white noise, and the per-capita growth rates of the competitors depend linearly on the resources, we prove that there is competitive exclusion. However, if either the dependence between the growth rates and the resources is not linear or the white noise term is nonlinear we show that coexistence on fewer resources than species is possible. Even more surprisingly, if the temporal environmental variation comes from switching the environment at random times between a finite number of possible states, it is possible for all species to coexist even if the growth rates depend linearly on the resources. We show in an example (a variant of which first appeared in Benaim and Lobry '16) that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.

Asymptotic harvesting of populations in random environments

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction-instead the harvested population converges to a unique invariant probability measure. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold [Formula: see text] such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is [Formula: see text] and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.