The optimal controlling strategy on a dispersing population in a two-patch system: Experimental and theoretical perspectives

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.

Resource allocation in a PDE ecosystem model

The effects of habitat heterogeneity on a diffusing population are investigated here. We formulate a reaction-diffusion system of partial differential equations to analyze the effect of resource allocation in an ecosystem with resource having its own dynamics in space and time. We show a priori estimates to prove the existence of state solutions given a control. We formulate an optimal control problem of our ecosystem model such that the abundance of a single species is maximized while minimizing the cost of inflow resource allocation. In addition, we show the existence and uniqueness of the optimal control as well as the optimal control characterization. We also establish the existence of an optimal intermediate diffusion rate. Moreover, we illustrate several numerical simulations with Dirichlet and Neumann boundary conditions with the space domain in 1D and 2D.