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.***