Diffusive Logistic Model Research Articles

Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity

In this paper, we study the Hopf bifurcation and spatiotemporal pattern formation of a delayed diffusive logistic model under Neumann boundary condition with spatial heterogeneity. It is shown that for large diffusion coefficient, a supercritical Hopf bifurcation occurs near the non-homogeneous positive steady state at a critical time delay value, and the dependence of corresponding spatiotemporal patterns on the heterogeneous resource function is demonstrated via numerical simulations. Moreover, it is proved that the heterogeneous resource supply contributes to the increase of the temporal average of total biomass of the population even though the total biomass oscillates periodically in time.

Maximization of the total population in a reaction–diffusion model with logistic growth

This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction–diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang–bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (Nonlinear Anal Real World Appl 11(2):688–704, 2010). To this purpose, we compute the first and second variations of the total population. When the growth rate is not of bang–bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. When the first variation becomes zero, we prove that the second variation is positive. These results implies that the bang–bang property is essential for the maximization of total population.

Partial differential equation modeling with Dirichlet boundary conditions on social networks

The being a wide range of applications of the Internet, social networks have become an effective and convenient platform for information communication, propagation and diffusion. Most of information exchange and spreading exist in social networks. The issue of information diffusion in social networks is getting more and more attention by government and individuals. The researchers investigated either empirical studies or focused on ordinary differential equation (ODE) models with only consideration of temporal dimension in most prior work. As is well known, partial differential equations (PDEs) can describe temporal and spatial patterns of information diffusion over online social networks; however, until now, results for understanding information propagation of social networks over both temporal and spatial dimensions are few. This paper is devoted to investigating a non-autonomous diffusive logistic model with Dirichlet boundary conditions to describe the process of information propagation in social networks. By constructing upper and lower solutions we obtain the dynamic behavior of the solution to the non-autonomous diffusive logistic model. Our results show that information diffusion is greatly affected by the diffusion coefficient d(t) and the intrinsic growth rate r(t).

The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment

This paper is concerned with a diffusive logistic model with advection and a free boundary in a spatially heterogeneous and time periodic environment. Such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front. Under more general assumptions on the initial data and the function standing for the intrinsic growth rate of the species, sharp criteria for spreading and vanishing are established, and estimates for spreading speed when spreading occurs are also derived. The obtained results considerably improve and complement the existing ones, especially those of [11,25].

A heterogeneous diffusive logistic model of a single species population dynamics with predation and harvesting terms

We study existence and multiplicity of positive solutions of a heterogeneous diffusive logistic equation with predation and harvesting terms, −Δu=au−b(x)u2−cu1+mu−dh(x)inΩ,∂u∂ν=0on∂Ω, where a,c,m and d are positive constants, Ω a bounded smooth domain in RN, and b(x) is a nonnegative function on Ω¯, with Ω0 a region such that Ω¯0⊂Ω and Ω¯0={x∈Ω:b(x)=0}. Under the strong growth rate assumption, that is, when a is greater than the first eigenvalue of −Δ in Ω0 with Dirichlet boundary condition, we show that the equation has at least one positive solution for 0≤d<da, for some da>0. In addition, in case c<a we prove some multiplicity results when m<m0, with m0=∫Ωb(x)dx/(a|Ω|).

The dynamics of a diffusive logistic model with nonlocal terms

It is well known that the set of positive solutions may contain crucial clues for the stationary patterns. In this paper, we consider a class of diffusive logistic equations with nonlocal terms subject to the Dirichlet boundary condition in a bounded domain. We study the existence of positive solutions under certain conditions on the parameters by using bifurcation theory. Finally, we illustrate the general results by applications to models with one-dimensional spatial domain.

Delay-induced periodic phenomenon in a diffusive regulated logistic model.

The diffusive logistic growth model with time delay and feedback control is considered. First, the well-posedness and permanence of solutions are discussed by using some comparison techniques. Then, the sufficient conditions for stability of nonnegative constant steady states are established, and the occurrence of Hopf bifurcation at positive steady state is performed. Next, the bifurcation properties are derived by computing the normal form on center manifold. Our results not only supplement but also generalized some existing ones. Finally, some numerical simulations show the feasibility of our theoretical analyses.

A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model

The spatial–temporal spreading of a new invasive species in a habitat has interest in ecology and is modeled by a moving boundary diffusion logistic partial differential problem, where the moving boundary represents the unknown expanding front of the species. In this paper a front-fixing approach is applied in order to transform the original moving boundary problem into a fixed boundary one. A finite difference method preserving qualitative properties of the theoretical solution is proposed. Results are illustrated with numerical experiments.

An Optimal Stopping Problem for Jump Diffusion Logistic Population Model

This paper examines an optimal stopping problem for the stochastic (Wiener-Poisson) jump diffusion logistic population model. We present an explicit solution to an optimal stopping problem of the stochastic (Wiener-Poisson) jump diffusion logistic population model by applying the smooth pasting technique (Dayanik and Karatzas, 2003; Dixit, 1993). We formulate this as an optimal stopping problem of maximizing the expected reward. We express the critical state of the optimal stopping region and the optimal value function explicitly.

A revisit to the diffusive logistic model with free boundary condition

This short paper revisits a free boundary problem which is used to describe the spreading of a new or invasive species. Our main goal is to understand how the underlying long-time dynamical behaviors response to the initial data. To this end, we parameterize the initial function as $u_0=\sigma\phi^*$, where $\sigma$ is regarded as a variable parameter and $\phi^*$ is a given function. Our main result suggests that when the diffusion rate is small, the species can persist in the long run (called spreading) for any $\sigma>0$; while if the diffusion rate is large, the species will go to extinction finally (called vanishing) for small $\sigma>0$. Maybe of more interest is that for some intermediate diffusion rates, there appears a sharp threshold value $\sigma^*\in(0, \infty)$ such that vanishing happens provided $0<\sigma\leq\sigma^*$ and spreading happens provided $\sigma>\sigma^*$. This result can be seen as an improvement of Theorem 1.2 in [8].

Stability and Bifurcation in a Diffusive Logistic Population Model with Multiple Delays

A diffusive logistic population model with multiple delays and Dirichlet boundary condition is considered in this paper. The stability/instability of the positive equilibrium and delay induced Hopf bifurcation are investigated. Moreover, we show which kind of delay could actually affect the dynamics.

A note on the numerical approach for the reaction–diffusion problem to model the density of the tumor growth dynamics

In this article, we numerically solve an equation modeling the evolution of the density of glioma in the brain—the most malignant form of brain tumor quantified in terms of net rates of proliferation and invasion. We employ a non-linear heterogeneous diffusion logistic density model. This model assumes that glioma cell invasion throughout the brain is a reaction–diffusion process and that the coefficient of diffusion can vary according to the gray and white matter composition of the brain at that location. The analysis provided in this article demonstrates that using the correct finite difference scheme can overcome the stability issues caused by the discontinuities of the diffusion coefficient. We also observe that at the steady-state these discontinuities vanish. To visualize and investigate numerically the behavior of the evolution of tumor concentration of the glioma, we calculated and plotted the number of tumor cells, the average mean radial distance, and the speed of the tumor cells along with charting the effects of net dispersal rate and net proliferation rate terms versus time for different center position values of Gaussian initial profile for each zone (gray and white matter tissues). We have proposed two numerical methods, the implicit backward Euler and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method for solving the governing equations. These methods are compared in terms of their performance in varying time-step and mesh-discretization. The Crank–Nicolson implicit method is shown to be the better choice to solve the equation.

Partial differential equations with Robin boundary condition in online social networks

In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, Digg.com. The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.

Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary

The structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined. In particular, a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary. Our existence results are established via the well-known sub-super solution method.

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 diffusive logistic model with a free boundary in time-periodic environment

We study the diffusive logistic equation with a free boundary in time-periodic environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For time independent environment, in the cases of one space dimension, and higher space dimensions with radial symmetry, this free boundary problem has been studied in Du and Lin (2010) [12], Du and Guo (2011) [9]. In both cases, a spreading–vanishing dichotomy was established, and when spreading occurs, the asymptotic spreading speed was determined. In this paper, we show that the spreading–vanishing dichotomy is retained in time-periodic environment, and we also determine the spreading speed. The former is achieved by further developing the earlier techniques, and the latter is proved by introducing new ideas and methods.

Erratum: Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free Boundary

Previous article Full AccessErratum: Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free BoundaryYihong Du and Zhigui LinYihong Du and Zhigui Linhttps://doi.org/10.1137/110822608PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractPropositions 4.1 and 4.3 in [Yihong Du and Zhigui Lin, SIAM J. Math. Anal., 42 (2010), pp. 377--405] contain some mistakes. In this erratum, we point out the correct versions of these results.1. G. Bunting, Y. Du and K. Krakowski , Spreading speed revisited: Analysis of a free boundary model , Netw. Heterog. Media , 7 ( 2012 ), pp. 583 -- 603 . CrossrefISIGoogle Scholar2. Y. Du and Z. Lin , Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary , SIAM J. Math. Anal. , 42 ( 2010 ), pp. 377 -- 405 . LinkISIGoogle Scholar3. Y. Du and Z. M. Guo , Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, II , J. Differential Equations , 250 ( 2011 ), pp. 4336 -- 4366 . CrossrefISIGoogle ScholarKeywordsdiffusive logistic equationfree boundaryspreading-vanishing dichotomyinvasive population Previous article FiguresRelatedReferencesCited byDetails A Free Boundary Problem of the Diffusive Competition Model with Different Habitats9 November 2021 | Journal of Dynamics and Differential Equations, Vol. 34, No. 3 Cross Ref Finite or Infinite Spreading Speed of an Epidemic Model with Free Boundary and Double Nonlocal Effects31 May 2022 | Journal of Dynamics and Differential Equations, Vol. 40 Cross Ref Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain7 December 2021 | Proceedings of the Edinburgh Mathematical Society, Vol. 65, No. 1 Cross Ref A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behaviorDiscrete and Continuous Dynamical Systems, Vol. 42, No. 6 Cross Ref Dynamics of West Nile virus driven by seasonal fluctuations in a spatially variable habitatDiscrete and Continuous Dynamical Systems - B, Vol. 0, No. 0 Cross Ref The diffusive intraguild predation model with intraspecific competition and double free boundaries23 January 2020 | Applicable Analysis, Vol. 100, No. 16 Cross Ref Dynamics for an Sir Epidemic Model with Nonlocal Diffusion and Free Boundaries1 June 2021 | Acta Mathematica Scientia, Vol. 41, No. 4 Cross Ref A Climate Shift Model with Free Boundary: Enhanced Invasion25 June 2021 | Journal of Dynamics and Differential Equations, Vol. 390 Cross Ref Competition in Periodic Media: III—Existence and Stability of Segregated Periodic Coexistence States11 February 2019 | Journal of Dynamics and Differential Equations, Vol. 32, No. 1 Cross Ref Instabilities in a combustion model with two free interfacesJournal of Differential Equations, Vol. 268, No. 7 Cross Ref Spreading and vanishing in a diffusive intraguild predation model with intraspecific competition and free boundary26 July 2019 | Mathematical Methods in the Applied Sciences, Vol. 42, No. 18 Cross Ref Biological invasion in a predator–prey model with a free boundary12 February 2019 | Boundary Value Problems, Vol. 2019, No. 1 Cross Ref Invasion of open space by two competitors: spreading properties of monostable two‐species competition‐diffusion systems27 May 2019 | Proceedings of the London Mathematical Society, Vol. 119, No. 5 Cross Ref Spreading speed for a West Nile virus model with free boundary23 April 2019 | Journal of Mathematical Biology, Vol. 79, No. 2 Cross Ref Competition in periodic media: II – Segregative limit of pulsating fronts and “Unity is not Strength”-type resultJournal of Differential Equations, Vol. 265, No. 1 Cross Ref A weak competition system with advection and free boundariesJournal of Mathematical Analysis and Applications, Vol. 463, No. 2 Cross Ref Using effective boundary conditions to model fast diffusion on a road in a large field15 September 2017 | Nonlinearity, Vol. 30, No. 10 Cross Ref A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries15 October 2015 | Journal of Dynamics and Differential Equations, Vol. 29, No. 3 Cross Ref A SIS reaction–diffusion–advection model in a low-risk and high-risk domainJournal of Differential Equations, Vol. 259, No. 10 Cross Ref Stefan problem, traveling fronts, and epidemic spreadDiscrete and Continuous Dynamical Systems - Series B, Vol. 21, No. 2 Cross Ref The spreading fronts in a mutualistic model with advectionDiscrete & Continuous Dynamical Systems - B, Vol. 20, No. 7 Cross Ref The spreading front of invasive species in favorable habitat or unfavorable habitatJournal of Differential Equations, Vol. 257, No. 1 Cross Ref Volume 45, Issue 3| 2013SIAM Journal on Mathematical Analysis History Submitted:28 January 2011Accepted:07 March 2013Published online:25 June 2013 Information© 2013, Society for Industrial and Applied MathematicsKeywordsdiffusive logistic equationfree boundaryspreading-vanishing dichotomyinvasive populationMSC codes35K2035R3535J6092B05PDF Download Article & Publication DataArticle DOI:10.1137/110822608Article page range:pp. 1995-1996ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

The diffusive logistic model with a free boundary and seasonal succession

This paper concerns a diffusive logistic equation with a freeboundary and seasonal succession, which is formulated to investigatethe spreading of a new or invasive species, where the free boundaryrepresents the expanding front and the time periodicity accounts forthe effect of the bad and good seasons. The condition to determinewhether the species spatially spreads to infinity or vanishes at afinite space interval is derived, and when the spreading happens,the asymptotic spreading speed of the species is also given. Theobtained results reveal the effect of seasonal succession on thedynamical behavior of the spreading of the single species.

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.

A diffusive logistic growth model to describe forest recovery

Land-use and land-cover change (LUCC) has broad implications for biodiversity, climate and ecosystem services. Even though LUCC often focuses on forest fragmentation, forest recovery is another form of LUCC that is becoming increasingly common. Understanding the process of forest recovery is a conservation and management priority; however, it is a difficult process to understand given the large number of factors that interact in a complex spatio-temporal setting. Reaction diffusion models provide an appropriate framework to study the complex dynamics of forest recovery because they account for both spatial structure and the dynamics of land-cover classes. Here, we describe a diffusive logistic growth (DLG) model to quantify forest recovery. We define a system in which forest diffuses through a non-forest matrix. The model consists of a diffusion term that describes the spread of forest in continuous space and time, and a logistic growth reaction that describes change in the proportion of forest. To illustrate model parameterization, we used the DLG approach to describe forest recovery in Puerto Rico from 1951 to 1991–1992. The model showed that forest recovery in Puerto Rico was explained by a positive intrinsic growth rate of forest and relatively slow diffusion. This mechanistic modeling approach presents a novel way to study forest recovery in continuous space and time while accounting for spatial dependency.