Stationary distributions of persistent ecological systems

Variation in environmental stochasticity dramatically affects viability and extinction time in a predator-prey system with high prey group cohesion

Understanding how tipping points arise is critical for population protection and ecosystem robustness. This work evaluates the impact of environmental stochasticity on the emergence of tipping points in a predator-prey system subject to the Allee effect and Holling type IV functional response, modeling an environment in which the prey has high group cohesion. We analyze the relationship between stochasticity and the probability and time that predator and prey populations in our model tip between different steady states. We evaluate the safety from extinction of different population values for each species, and accordingly assign extinction warning levels to these population values. Our analysis suggests that the effects of environmental stochasticity on tipping phenomena are scenario-dependent but follow a few interpretable trends. The probability of tipping towards a steady state in which one or both species go extinct generally monotonically increased with noise intensity, while the probability of tipping towards a more favorable steady state (in which more species were viable) usually peaked at intermediate noise intensity. For tipping between two equilibria where a given species was at risk of extinction in one equilibrium but not the other, noise affecting that species had greater impact on tipping probability than noise affecting the other species. Noise in the predator population facilitated quicker tipping to extinction equilibria, whereas prey noise instead often slowed down extinction. Changes in warning level for initial population values due to noise were most apparent near attraction basin boundaries, but noise of sufficient magnitude (especially in the predator population) could alter risk even far away from these boundaries. Our model provides critical theoretical insights for the conservation of population diversity: management criteria and early warning signals can be developed based on our results to keep populations away from destructive critical thresholds.

Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity

May and Leonard (SIAM J Appl Math 29:243-253, 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each "cycle", passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic Escherichia coli), in the immune system, in neural information processing models ("winnerless competition"), and in models of neural central pattern generators. Yet as May and Leonard observed "Biologically, the behavior (produced by the model) is nonsense. Once it is conceded that the variables represent animals, and therefore cannot fall below unity, it is clear that the system will, after a few cycles, converge on some single population, extinguishing the other two." Here, we explore different ways of introducing discrete stochastic dynamics based on May and Leonard's ODE model, with application to ecological population dynamics, and to a neuromotor central pattern generator system. We study examples of several quantitatively distinct asymptotic behaviors, including total extinction of all species, extinction to a single species, and persistent cyclic dominance with finite mean cycle length.

The effects of random and seasonal environmental fluctuations on optimal harvesting and stocking

We analyze the harvesting and stocking of a population that is affected by random and seasonal environmental fluctuations. The main novelty comes from having three layers of environmental fluctuations. The first layer is due to the environment switching at random times between different environmental states. This is similar to having sudden environmental changes or catastrophes. The second layer is due to seasonal variation, where there is a significant change in the dynamics between seasons. Finally, the third layer is due to the constant presence of environmental stochasticity-between the seasonal or random regime switches, the species is affected by fluctuations which can be modelled by white noise. This framework is more realistic because it can capture both significant random and deterministic environmental shifts as well as small and frequent fluctuations in abiotic factors. Our framework also allows for the price or cost of harvesting to change deterministically and stochastically, something that is more realistic from an economic point of view. The combined effects of seasonal and random fluctuations make it impossible to find the optimal harvesting-stocking strategy analytically. We get around this roadblock by developing rigorous numerical approximations and proving that they converge to the optimal harvesting-stocking strategy. We apply our methods to multiple population models and explore how prices, or costs, and environmental fluctuations influence the optimal harvesting-stocking strategy. We show that in many situations the optimal way of harvesting and stocking is not of threshold type.