Global dynamics and spatial patterns of a ratio-dependent preytaxis model driven by the acceleration

Abstract

The movement of individual organism has been recognized for a long time as a significant factor influencing the spatio-temporal distribution of populations. Numerous reaction-diffusion models that can track spatial and temporal changes in population size have been created to explain how the movement of individuals affects the spatial and temporal distribution of biological populations. Using these reaction-diffusion systems, variety of biological processes, such as reproduction or genetics, tumor growth, wound healing, patch production, etc., have been demonstrated. In these reaction-diffusion models, the dispersal strategy of individuals is typically assumed to be random diffusion; however, it cannot explain some of the more complex ecological processes involving rational movements (e.g., directed movements of dispersing individuals that are chemotaxis, preytaxis, etc. to increase their chances of survival) nor accurately reflect the non-Brownian movements of individuals. If diffusion is assumed to be only random, no spatially inhomogeneous patterns will be observed for the predator-prey system, which cannot explain the spatio-temporal heterogeneity of patterns observed in the experiment. Therefore, it is more reasonable to incorporate rational motion into the model in certain real-world circumstances. In this thesis, we study the celebrated predator-prey systems with preytaxis, where the taxis term represting the rational movement is formulated as an advection term. Numerous investigations addressing predator-prey interactions have demonstrated that in some preytaxis models, it is more plausible to assume that the predator’s acceleration (rather than preytactic velocity) is proportional to the prey density gradient. Such acceleration-driven preytaxis models were introduced in [9, 39] to explain the observed spatial heterogeneity of predators and prey. This thesis is dedicated to exploring the global dynamics of a ratio-dependent preytaxis system driven by acceleration. The existence of classical solutions with uniform-in-time bound was established in any spatial dimension. Moreover, we prove the global stability of the spatially homogeneous prey-only and coexistence steady states under certain conditions on system parameters and show that the convergence rates are exponential type. For the system parameters outside the stability regime, linear stability analysis is performed to find the possible patterning regimes and numerical simulations are used to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise which can interpret the spatial-temporal heterogeneity observed in experiments.