Modeling and analysis of some biological models with cross-diffusion

Abstract

Cross-diffusion, a process in which the density gradient of one species induces an advective flux of another species, has been widely applied to model the movement of one species toward or away from the area with higher density of another species (i.e., taxis movement). In this thesis, we first study an indirect prey-taxis model [129] with anti-predation mechanism. Then, we explore another type of anti-predation mechanism: alarm-taxis [46] which described by a three-species Lotka-Volterra food chain model. Next, we apply the cross-diffusion strategy to an SIS epidemic model and numerically explored the effects of cross-diffusion. Finally, we investigate a toxicant-taxis model and theoretically prove the effects of cross-diffusion. Fundamentally, we establish the global boundedness of classical solutions by using energy estimates. The other main results are as follows: 1. For the indirect prey-taxis model with anti-predation, we prove the global stability of constant steady states by constructing energy functionals. Moreover, when the prey adopts the anti-predation strategy, we establish the existence of non-constant positive steady-state solutions by applying Leray-Schauder degree theory and prove that no Hopf bifurcation occurs. These pattern formation results are different from both indirect prey-taxis (which exhibits Hopf bifurcation) and the case without cross-diffusion (where no patterns emerge). 2. For the three-species Lotka-Volterra food chain model with intraguild predation and taxis mechanisms, we build the global stability of constant steady states by using energy functionals. Moreover, we numerically demonstrate that the combination of taxis mechanisms and intraguild predation can induce rich pattern formations. Notably, our simulations show that prey-taxis may have a destabilizing effect in food chain model with intraguild predation, which contrasts with the well-known stabilizing effect observed in two-species predator-prey systems or the food chain model with alarm-taxis but without intraguild predation. 3. For the SIS model with a cross-diffusion dispersal strategy for the infected individuals, which describes the public health intervention measures, we define the basic reproduction number R0. Then we employ a change of variable and apply the index theory along with the principal eigenvalue theory to establish the threshold dynamics in terms of R0. Moreover, we explore the global stability of constant steady states. Finally, we numerically demonstrate that the cross-diffusion strategy can reduce R0 and help eradicate the diseases even if the habitat is high-risk in contrast to the situation without cross-diffusion. 4. For the toxicant-taxis model in a time-periodic environment, we prove the existence of positive periodic solutions and the uniform persistence by applying the uniform persistence theory and Principal Floquet bundle theory. Moreover, we establish the global stability of non-constant periodic solutions through energy methods. By studying the effects of the strong toxicant-taxis on the corresponding periodic principal eigenvalue, we theoretically prove that the strong toxicant-taxis (i.e., cross-diffusion) helps aquatic species to survive. Moreover, we develop new ideas to overcome the difficulties caused by the failure of the comparison principle in cross-diffusion models. For example, the proof ideas developed in Chapter 5 can be applied to prove the existence of time-periodic/non-constant steady-state solutions, and uniform persistence for general cross-diffusion models.