On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains

We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.     

Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect

A reaction-diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel.     

Numerical solutions of diffusive logistic equation

In this paper we investigate numerically positive solutions of a superlinear Elliptic equation on bounded domains.The study of Diffusive logistic equation continues to be an active field of research. The subject has important applications to population migration as well as many other branches of science and engineering. In this paper the “finite difference scheme” will be developed and compared for solving the one- and three-dimensional Diffusive logistic equation. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from many authors these years.     

Exact multiplicity of solutions to a diffusive logistic equation with harvesting

An Ambrosetti-Prodi type exact multiplicity result is proved for a diffusive logistic equation with harvesting. We show that a modified diffusive logistic mapping has exactly either zero, or one, or two pre-images depending on the harvesting rate. It implies that the original diffusive logistic equation with harvesting has at most two positive steady state solutions.     

Optimal harvesting for periodic age-dependent population dynamics with logistic term

We prove an asymptotic behavior result for an age-dependent population dynamics with logistic term and periodic vital rates. We investigate next an optimal harvesting problem related to a periodic age-structured model with logistic term. Existence of an optimal control and necessary optimality conditions are established. A conceptual algorithm to approximate the optimal pair is derived and some numerical experiments are presented.     

Stochastic differential delay equations of population dynamics

In this paper we stochastically perturb the delay Lotka-Volterra model

     

A finite difference approach to the equations of age-dependent population dynamics

We establish the convergence of the finite difference scheme for the nonlinear equations of population dynamics proposed by Guertin and MacCamy. The applicability of the discrete equations to establish qualitative properties of the solution to the continuous problem is also illustrated.     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

A numerical method for finding positive solution of diffusive logistic equation

Using a numerical methods based on sub–super solution, we will find positive solution for the diffusive logistic equation Δu+au-bu²=0 for xΩ, with Dirichlet boundary condition.     

A finite difference scheme for the equations of population dynamics

In this paper, we develop a discrete age-dependent population model by applying a finite difference scheme to the McKendrick type population equation. The properties of the system are analyzed. The convergence of the sequence of solutions and the critical fertility to those of the continuous model are proved (as the age step approaches zero).     

Symmetry,conserved quantities and moments in diffusive equations

In this note we show that there is an intimate connection between the symmetries of a partial differential equation, and the time behavior of its moments. We also derive closed-form expressions for certain conserved quantities and polynomial solutions. We discuss the heat and Fokker-Planck equations.     

Periodic solutions of nonlinear equations generalizing logistic equations with delay

Mathematical Notes -     

Survival to extinction in a slowly varying harvested logistic population model

This work considers a harvested logistic population for which birth rate, carrying capacity and harvesting rate all vary slowly with time. Asymptotic results from earlier work, obtained using a multiscaling technique, are combined to construct approximate expressions for the evolving population for the situation where the population initially survives to a slowly varying limiting state, but then, due to increasing harvesting, is reduced to extinction in finite time. These results are shown to give very good agreement with those obtained from numerical computation.     

Nonlinear age-dependent diffusive equations: A bifurcation approach

The main goal of this paper is the study of the existence and uniqueness of positive solutions of some nonlinear age-dependent diffusive models, arising from dynamic populations. We use a bifurcation method, for which it has been necessary to study in detail the linear and eigenvalue problems associated to the nonlinear problem in an appropriate space.