Global dynamics of some predator-prey systems with preytaxis

Abstract

Organisms cannot live without food resource as their energy supply, in all probability. The different strategies that they use to forage or to increase their survival rates may result in diverse interactions between or among organisms, amongst which predation as one of fundamental relations exists broadly in nature. This thesis is associated with exploring dynamics of classical solution to two classes of predator-prey models with spatial diffusion and preytaxis effect: direct preytaxis and indirect preytaxis. The preytaxis here refers to that predators have an apparent tendency to move towards the region of higher density of prey. The main difference of being direct or indirect case lies in that predators search for prey directly, or perceive mainly the signals released by prey through which predators may likely find the prey eventually. In more detail, our results include three parts as below: Firstly, for the direct preytaxis model with no diffusion of prey (i.e., a parabolic-ODE system), we study local-in-time existence and uniqueness of its classical solution by using Banach's fixed-point theory in a suitable Sobolev space as the spatial domain Ω C Rn(n ≥ 1). Also, we derive its global existence by obtaining uniform-in-time boundedness of its solution in norm L∞(Ω), when spatial dimension n = 2. On the other hand, inspired by vanishing viscosity method we explore convergence relationship between the strong solution of a related fully parabolic PDE system and the aforementioned parabolic-ODE system in Ω C R² , when the diffusion coefficient ε (> 0) of prey density tends to zero. Here the main tools used include analytic semigroup techniques, Aubin-Lions compactness lemma, trace interpolation inequalities, Lp theory and Schauder's estimate of linear parabolic equations, etc. Finally, for the indirect preytaxis model with density-dependent preytaxis we investigate global-in-time existence, uniqueness and uniform-in-time boundedness of its classical solution in Ω C Rn (n ≥ 1), by a combination of Amann's theory for quasi-linear parabolic systems, analytical semigroup techniques and Moser's iteration. In addition, via Lyapunov's function techniques and limit property of dynamical systems we acquire that the classical solution may converge in norm L∞(Ω), as time t → +∞, to its prey-only state and coexistence state under suitable conditions. The numerical simulations we perform indicate that some density-dependent preytaxis and predators' diffusion may either flatten the spatial one-dimensional patterns which exist in non-density-dependent case, or break the spatial two-dimensional distribution similarity which occurs in non-density-dependent case between predators and chemoattractants (released by prey).