Existence, Multiplicity And Uniqueness Of Positive Steady-states For Biological Models

Many ecological phenomena in nature can be rationalized into mathematical models. By investigating these models, some ecological phenomena may be ex-plained and controlled scientifically, and some reasonable schemes may be provided for the solution of ecological problems.As early as the beginning of the nineteenth century, mathematicians have made use of ordinary differential equations to describe the evolution of biological popula-tions. During that period, the issues discussed by people were under the assumption that the population density in space distribution is uniform. If the density distribu-tion is non-uniform, then high-density populations would diffuse to the position of low-density. At this point, a large number of bio-mathematical models can be sum-marized as reaction-diffusion equations. Using reaction-diffusion systems to study the dynamical behavior of biological populations has been an important research as-pect in the region of nonlinear partial differential equations. Since the population's long time behavior is closely related to the steady-state problem of reaction-diffusion system, studies on the qualitative properties of positive steady-state solutions have an important significance both in theory and in reality.In this dissertation, mainly using the theories of nonlinear analysis and nonlin-ear partial differential equations, especially those of reaction-diffusion equations and the corresponding elliptic equations, we have systematically studied the dynamical behaviors of four biological models with reaction and diffusion, such as the coex-istence, multiplicity, uniqueness and stability of positive steady-states. The tools used here include super-sub solutions method, comparison principle, global bifurca-tion theory, fixed-point theory of topology and perturbation technique. The main contents and results are as follows:(i) A diffusive predator-prey model with Beddington-DeAngelis functional re-sponse under homogeneous Dirichlet boundary conditions is studied once more. Making use of global bifurcation theory, we obtain a necessary and sufficient con-dition for the existence of positive solutions. Moreover, a range of parameters for the uniqueness of positive solution is described in one dimension. Furthermore, the effect of large k which represents the extent of mutual interference between preda-tors is extensively studied. By meticulously analyzing the asymptotic behaviors, we obtain a complete understanding of the existence, uniqueness and stability of positive solutions when k is sufficiently large.(ii) We consider a reaction-diffusion system of three species:predator, prey and mutualist, and investigate the multiplicity and uniqueness of positive steady-states. Here we consider a case in which a mutualist modifies predation to the benefit of a prey, namely a mutualist deterring predation on a prey. By means of the index theory of fixed points, we obtain two sufficient conditions for the existence of positive solutions, and whenβis suitably large, we establish the multiplicity result and find all the positive solutions are of only two types, one is asymptotically stable and the other unstable. Moreover, by the classic regular and singular perturbation theory, we extensively study the effect ofγ, which represents the extent of a mutualist deterring predation on a prey, and gain a good understanding of the existence, uniqueness and stability of positive solutions whenγis sufficiently large. Our further results show that the uniqueness does not necessarily needγto be large at some moment, and we establish a more general result even whenγis bounded.(iii) A cross-diffusion predator-prey model with modified Leslie-Gower and Holling-II functional responses is discussed. Making use of global bifurcation the-ory, we obtain two sufficient conditions for the existence of positive solutions and then describe the coexistence region. Moreover, by analyzing the cross-diffusion co-efficients dependence of two boundary curves, we find that the coexistence region spreads as the effect of prey on predator's diffusion increases, and narrows when the effect of predator on prey's diffusion is large. At last, we derive the effect of nonlin-ear diffusion on positive solutions. The results show that all the positive solutions have only two possible asymptotic behaviors.(iv) A competition model in the unstirred chemostat is considered. The bifur-cation solution (u(s),v(s)) from a double eigenvalue is obtained. And we see that for sufficiently small s> 0, (u(s),v(s)) connects the bifurcating positive solution from the semitrivial solution (θa,0) with that from the other semitrivial solution (0,θb). Moreover, the asymptotic stability of (u(s),v(s)) is derived under certain conditions.