Diffusive Logistic Model Research Articles

Radial Basis Function–Finite Difference Solution Combined with Level-Set Embedded Boundary Method for Improving a Diffusive Logistic Model with a Free Boundary

In this paper, we propose an efficient numerical method to solve the problems of diffusive logistic models with free boundaries, which are often used to simulate the spreading of new or invasive species. The boundary movement is tracked by the level-set method, where the Hamilton–Jacobi weighted essentially nonoscillatory (HJ-WENO) scheme is utilized to capture the boundary curve embedded by the Cartesian grids via the embedded boundary method. Then the radial basis function–finite difference (RBF-FD) method is adopted for spatial discretization and the implicit–explicit (IMEX) scheme is considered for time integration. A variety of numerical examples are utilized to demonstrate the evolution of the diffusive logistic model with different initial boundaries.

A random free-boundary diffusive logistic differential model: Numerical analysis, computing and simulation

A free boundary diffusive logistic model finds application in many different fields from biological invasion to wildfire propagation. However, many of these processes show a random nature and contain uncertainties in the parameters. In this paper we extend the diffusive logistic model with unknown moving front to the random scenario by assuming that the involved parameters have a finite degree of randomness. The resulting mathematical model becomes a random free boundary partial differential problem and it is addressed numerically combining the finite difference method with two approaches for the treatment of the moving front. Firstly, we propose a front-fixing transformation, reshaping the original random free boundary domain into a fixed deterministic one. A second approach is using the front-tracking method to capture the evolution of the moving front adapted to the random framework. Statistical moments of the approximating solution stochastic process and the stochastic moving boundary solution are calculated by the Monte Carlo technique. Qualitative numerical analysis establishes the stability and positivity conditions. Numerical examples are provided to compare both approaches, study the spreading-vanishing dichotomy, prove qualitative properties of the schemes and show the numerical convergence.

Numerical Algorithm for Source Determination in a Diffusion–Logistic Model from Integral Data Based on Tensor Optimization

Numerical Algorithm for Source Determination in a Diffusion–Logistic Model from Integral Data Based on Tensor Optimization

Numerical Algorithm for Source Determination in a Diffusion–Logistic Model from Integral Data Based on Tensor Optimization

An algorithm has been developed for numerically solving the source determination problem in the model of information dissemination in synthetic online social networks, described by reaction–diffusion-type equations, using additional information about the process at fixed time points. The degree of ill-posedness of the source determination problem for a parabolic equation is studied based on the analysis of singular values of the linearized operator of the inverse problem. The algorithm developed is based on a combination of the tensor optimization method and gradient descent supplemented with the A.N. Tikhonov regularization. Numerical calculations demonstrate the smallest relative error in the reconstructed source obtained by the developed algorithm in comparison with classical approaches.

Stability and bifurcation in a reaction–diffusion model with nonlinear boundary conditions

In this paper, we investigate the existence, stability, local and global bifurcation of the steady state solutions for a diffusive logistic model with nonlinear boundary conditions. Supercritical and subcritical bifurcation of steady state solutions are obtained by virtue of the Crandall–Rabinowitz theorem and the Lyapunov–Schmidt reduction. The local stability and the possibility of obtaining an Allee effect are also analyzed. Furthermore, the result on global bifurcation is obtained as well by employing the Rabinowitz theorem.

Qualitative Numerical Analysis of a Free-Boundary Diffusive Logistic Model

A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary can be treated by a front-fixing approach resulting in a fixed-domain non-linear problem, which is solved by an explicit finite difference method. Qualitative numerical analysis establishes the stability, positivity and monotonicity conditions. Special attention is paid to the spreading–vanishing dichotomy and a numerical algorithm for the spreading–vanishing boundary is proposed. Theoretical statements are illustrated by numerical tests.

Maximal total population of species in a diffusive logistic model.

In this paper, we investigate the maximization of the total population of a single species which is governed by a stationary diffusive logistic equation with a fixed amount of resources. For large diffusivity, qualitative properties of the maximizers like symmetry will be addressed. Our results are in line with previous findings which assert that for large diffusion, concentrated resources are favorable for maximizing the total population. Then, an optimality condition for the maximizer is derived based upon rearrangement theory. We develop an efficient numerical algorithm applicable to domains with different geometries in order to compute the maximizer. It is established that the algorithm is convergent. Our numerical simulations give a real insight into the qualitative properties of the maximizer and also lead us to some conjectures about the maximizer.

Comparative Analysis of Gradient Methods for Source Identification in a Diffusion-Logistic Model

Comparative Analysis of Gradient Methods for Source Identification in a Diffusion-Logistic Model

A heterogeneous diffusive logistic model with constant yield harvesting in $\mathbb{R^N}$ under strong growth rate

A heterogeneous diffusive logistic model with constant yield harvesting in $\mathbb{R^N}$ under strong growth rate

Global attractivity of delayed and nonlocal diffusive logistic models with variable coefficients

In this paper, we consider a reaction-diffusion equation with continuous delay and spatial variable coefficients which models the evolution of a single species. We establish a sharp threshold dynamic result: there exists a critical value λ ( a ) such that if λ ( a ) < 0 the positive steady state solution of the equation is globally attractive, while if λ ( a ) ≥ 0 the trivial steady state is globally attractive. To this end, we analyze the ω -limit set of the equation and prove that it is a singleton. Moreover, we apply our method to obtain global attractivity of the positive steady state of a spatially nonlocal diffusive logistic model.

Effects of impulsive harvesting and an evolving domain in a diffusive logistic model*

In order to understand how the combination of the evolution of a domain and impulsive harvesting affect the dynamics of a population, we investigate a diffusive logistic population model with impulsive harvesting on a periodically evolving domain. Initially the ecological reproduction index of the impulsive problem is introduced and given by an explicit formula, which depends on the domain evolution rate and the impulsive function. Then the threshold dynamics of the population subject to monotone or nonmonotone impulsive harvesting are established based on this index. Finally numerical simulations are carried out to illustrate our theoretical results, and these reveal that a large domain evolution rate can improve the populations ability to survive, no matter which impulsive harvesting takes place. On the contrary, impulsive harvesting has a negative effect on the survival of the population, and can even lead to its extinction.

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.

New alternative numerical approaches for solving the glioma model and their efficiencies

In this article, the numerical solution of glioma, or glioblastomas, which is one of the most aggressive forms of cancer is considered. A heterogeneous nonlinear diffusion logistic density model is taken as the main focus. To obtain the numerical results, three different discretization techniques: Pseudospectral method (PSM) using Chebyshev–Gauss–Lobatto collocation points, method of lines (MoL), and cubic B-splines (cBS) are employed on the spatial domain, whereas 4th-order Runge–Kutta (RK4) is considered on the time domain. Adapting cBS and PSM discretization to the glioma model is studied at first in this study. In addition to the theoretical convergence results, detailed comparative computational results are presented. All these methods are compared in terms of their efficiencies in varying time step and mesh discretization not only to one another, but also with the methods given in the literature.

Memory-based movement with spatiotemporal distributed delays in diffusion and reaction

In this paper, we investigate the spatiotemporal dynamics of a single-species model with spatiotemporal delays characterizing spatial memory and maturation. Through stability and bifurcation analysis, we find that the spatial memory-based diffusion coefficient, the spatiotemporal diffusive delay and spatiotemporal reaction delay have important effects on the dynamics of the model and their combined impact can cause the destabilization of the positive constant steady state and give rise to steady state and Hopf bifurcations. Taking the coefficient of spatial memory diffusion as the bifurcation parameter, the critical values of steady state and Hopf bifurcations are rigorously determined. Furthermore, we apply the theoretical results to a modified diffusive logistic model with predation and obtain spatially inhomogeneous steady states and spatially homogeneous and inhomogeneous periodic solutions via numerical simulations.

Global and local optimization in identification of parabolic systems

Abstract The problem of identification of coefficients and initial conditions for a boundary value problem for parabolic equations that reduces to a minimization problem of a misfit function is investigated. Firstly, the tensor train decomposition approach is presented as a global convergence algorithm. The idea of the proposed method is to extract the tensor structure of the optimized functional and use it for multidimensional optimization problems. Secondly, for the refinement of the unknown parameters, three local optimization approaches are implemented and compared: Nelder–Mead simplex method, gradient method of minimum errors, adaptive gradient method. For gradient methods, the evident formula for the continuous gradient of the misfit function is obtained. The identification problem for the diffusive logistic mathematical model which can be applied to social sciences (online social networks), economy (spatial Solow model) and epidemiology (coronavirus COVID-19, HIV, etc.) is considered. The numerical results for information propagation in online social network are presented and discussed.

Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment

In this paper we study a nonlocal diffusion model with double free boundaries in time periodic environment, which is the natural extension of the free boundary model in [ 17 ], where local diffusion is used to describe the population dispersal. We give the existence and uniqueness of global solution and consider the properties of principle eigenvalue of time-periodic parabolic-type eigenvalue problem. With the help of attractivity of time periodic solutions, we establish a spreading-vanishing dichotomy. The sharp criteria for spreading and vanishing are also obtained.

A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate

In this article, we discuss a space-fractional diffusion logistic population model with Caputo fractional derivative and density-dependent dispersal rate. The numerical solution of the problem is obtained by using a finite difference scheme. The consistency and stability of the scheme for our solution to the problem are also discussed. The effect of the density-dependent dispersal rate and order of the space-fractional derivative are analyzed for the population density and expanding front (moving boundary).

On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments.

We first consider a diffusive logistic model of a single species in a heterogeneous environment, with two parameters, r(x) for intrinsic growth rate and K(x) for carrying capacity. When r(x) and K(x) are proportional, i.e., [Formula: see text], it is proved by Lou (J Differ Equ 223(2):400-426, 2006) that a population diffusing at any rate will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This paper studies another case when r(x) is a constant, i.e., independent of K(x). In such case, a striking result is that for any dispersal rate, the logistic equation with spatially heterogeneous resources will always support a total population strictly smaller than the total carrying capacity at equilibrium, which is just opposite to the case [Formula: see text]. These two cases of single species models also lead to two different forms of Lotka-Volterra competition-diffusion systems. We then examine the consequences of the aforementioned difference on the two forms of competition systems. We find that the outcome of the competition in terms of the dispersal rates and spatial distributions of resources for the two forms of competition systems are again quite different. Our results indicate that in heterogeneous environments, the correlation between r(x) and K(x) has more profound impacts in population ecology than we had previously expected, at least from a mathematical point of view.

On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model

We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely, \begin{document}$ \begin{equation*} \label{abs} \left\lbrace \begin{matrix}-\Delta u = \lambda u(1-u) ;\; x\in\Omega \frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u = 0; \; x\in\partial \Omega \end{matrix} \right. \end{equation*} $\end{document} where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^N $\end{document} ; \begin{document}$ N > 1 $\end{document} with smooth boundary \begin{document}$ \partial \Omega $\end{document} or \begin{document}$ \Omega = (0,1) $\end{document} , \begin{document}$ \frac{\partial u}{\partial \eta} $\end{document} is the outward normal derivative of \begin{document}$ u $\end{document} on \begin{document}$ \partial \Omega $\end{document} , \begin{document}$ \lambda $\end{document} is a domain scaling parameter, \begin{document}$ \gamma $\end{document} is a measure of the exterior matrix ( \begin{document}$ \Omega^c $\end{document} ) hostility, and \begin{document}$ A\in (0,1) $\end{document} and \begin{document}$ \epsilon>0 $\end{document} are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for \begin{document}$ u and increasing for \begin{document}$ u>A $\end{document} . We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of \begin{document}$ \lambda $\end{document} . When \begin{document}$ \Omega = (0,1) $\end{document} we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter \begin{document}$ \gamma $\end{document} varies. Our results indicate that when \begin{document}$ \gamma $\end{document} is large there is no Allee effect for any \begin{document}$ \lambda $\end{document} . We employ a method of sub-supersolutions to obtain existence and multiplicity results when \begin{document}$ N>1 $\end{document} , and the quadrature method to study the case \begin{document}$ N = 1 $\end{document} .

Dirichlet problem for a diffusive logistic population model with two delays

In this paper, we investigate a diffusive logistic equation with non-zero Dirichlet boundary condition and two delays. We first exclude the existence of positive heterogeneous steady states, which implies the uniqueness of constant positive steady state. Then, we analyze the local stability and local Hopf bifurcation at the positive steady state. We show that multiple delays can induce multiple stability switches. Furthermore, we prove global stability of the positive steady state under certain conditions and obtain global Hopf bifurcation results. Our theoretical results are illustrated with numerical simulations.