Systems with a distributed memory and applications to population dynamics

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The main focus of the proposed research is on the interplay of continuous systems with time delay and discrete systems. It is well known that a nonlinear equation with a constant delay in the nonlinear production term and a linear mortality term is stable for any delay, if the discrete map with the same nonlinear function is stable. We have recently generalized this result to a scalar non-autonomous equation with a distributed delay in the production term only. However, the result is no longer true for systems of delay differential equations, and a more complicated criterion must be applied. The purpose of the proposed research is to extend these results to systems with a distributed variable delay. We plan to consider distributed delays of two types: finite (but not necessarily bounded) and infinite but with exponentially decaying memory.   For discrete models, the proposed research will target to analyze cyclic rather than stable dynamics and the cases when stochastic perturbations can significantly change the dynamics of the system. As an example, we will consider systems of population dynamics with the Allee effect (extinction for small enough population levels) where either the Allee effect can be alleviated by applying a stochastic perturbation with zero mean, or there is a significant probability of extinction even for large initial population values, due to stochastic perturbations. Deterministic and stochastic control of otherwise unstable discrete systems will also be one of the topics of the proposed research. Recently significant progress has been achieved on stabilization of otherwise unstable maps. For example, the target oriented control introduced in 2011 allows to stabilize a wide class of one-dimensional maps, even giving control over the choice of a stable equilibrium point. In the case of pulse control applied not at every step, a cycle can be stabilized. Extending these results to higher order difference equations and systems is one of the purposes of the present research. Another important aspect of the research is the influence of different diffusion strategies on the success of competing spatially distributed species, where the model is described by a system of partial differential equations. So far we have considered systems of two species, where the only difference between them is the diffusion strategy. We plan to extend these results to Lotka-Volterra competition models and explore the interplay of the following factors: diffusion strategy, efficiency of resource exploitation (expressed in higher or lower carrying capacity) and intraspecies interactions.