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Discrete and Continuous Dynamical Systems

June 2020 , Volume 40 , Issue 6

DCDS-A special issue to honor Wei-Ming Ni's 70th birthday

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Preface: DCDS-A special issue to honor Wei-Ming Ni's 70th birthday
Chiun-Chuan Chen, Yuan Lou, Hirokazu Ninomiya, Peter Polacik and Xuefeng Wang
2020, 40(6): i-ii doi: 10.3934/dcds.2020171 +[Abstract](2653) +[HTML](622) +[PDF](78.36KB)
Bifurcation from infinity with applications to reaction-diffusion systems
Chihiro Aida, Chao-Nien Chen, Kousuke Kuto and Hirokazu Ninomiya
2020, 40(6): 3031-3055 doi: 10.3934/dcds.2020053 +[Abstract](2240) +[HTML](530) +[PDF](380.03KB)

The bifurcation method is one of powerful tools to study the existence of a continuous branch of solutions. However without further analysis, the local theory only ensures the existence of solutions within a small neighborhood of bifurcation point. In this paper we extend the theory of bifurcation from infinity, initiated by Rabinowitz [11] and Stuart [13], to find solutions of elliptic partial differential equations with large amplitude. For the applications to the reaction-diffusion systems, we are able to relax the conditions to obtain the bifurcation from infinity for the following nonlinear terms; (ⅰ) nonlinear terms satisfying conditions similar to [11] (all directions), (ⅱ) nonlinear terms satisfying similar conditions only on the strip domain along the direction determined by the eigenfunction, (ⅲ) \begin{document}$ p $\end{document}-homogeneous nonlinear terms with degenerate conditions.

Evolution equations involving nonlinear truncated Laplacian operators
Matthieu Alfaro and Isabeau Birindelli
2020, 40(6): 3057-3073 doi: 10.3934/dcds.2020046 +[Abstract](1636) +[HTML](510) +[PDF](378.54KB)

We first study the so-called Heat equation with two families of elliptic operators which are fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equation with operators including the "large" eigenvalues has strong similarities with a Heat equation in lower dimension whereas, surprisingly, for operators including "small" eigenvalues it shares some properties with some transport equations. In particular, for these operators, the Heat equation (which is nonlinear) not only does not have the property that "disturbances propagate with infinite speed" but may lead to quenching in finite time. Last, based on our analysis of the Heat equations (for which we provide a large variety of special solutions) for these operators, we inquire on the associated Fujita blow-up phenomena.

Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small
Xueli Bai and Fang Li
2020, 40(6): 3075-3092 doi: 10.3934/dcds.2020035 +[Abstract](1863) +[HTML](599) +[PDF](384.66KB)

In this paper, we study the global dynamics of a general \begin{document}$ 2\times 2 $\end{document} competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper extends the work in [3], where Lotka-Volterra competition models with symmetric nonlocal operators are considered, to more general competition models with nonsymmetric operators.

Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar and Wen Yang
2020, 40(6): 3093-3116 doi: 10.3934/dcds.2020039 +[Abstract](2033) +[HTML](501) +[PDF](475.57KB)

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on an arbitrary flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus.

A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation
Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura and Tohru Wakasa
2020, 40(6): 3117-3142 doi: 10.3934/dcds.2019226 +[Abstract](3828) +[HTML](927) +[PDF](496.79KB)

We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.

Signed Radon measure-valued solutions of flux saturated scalar conservation laws
Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina and Alberto Tesei
2020, 40(6): 3143-3169 doi: 10.3934/dcds.2020041 +[Abstract](2097) +[HTML](503) +[PDF](496.87KB)

We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.

On the spectral theory of positive operators and PDE applications
Kung-Ching Chang, Xuefeng Wang and Xie Wu
2020, 40(6): 3171-3200 doi: 10.3934/dcds.2020054 +[Abstract](3091) +[HTML](968) +[PDF](460.48KB)

The strong version of the Krein-Rutman theorem requires that the positive cone of the Banach space has nonempty interior and the compact map is strongly positive, mapping nonzero points in the cone into its interior. In this paper, we first generalize this version of the Krein-Rutman theorem to the case of "semi-strongly positive" operators; and we prove it in a totally elementary fashion. We then prove the equivalence of semi-strong positivity and irreducibility in a Banach lattice, linking the afore-mentioned result with the Krein-Rutman theorem for irreducible operators. One of the things we emphasize is to use "upper and lower spectral radii" to characterize, in the fashion of Collatz-Wielandt formula for nonnegative irreducible matrices, the principal eigenvalue of these operators. For reducible operators, we prove that the lower spectral radius always serves as the least upper bound of the set of eigenvalues pertaining to positive eigenvectors, and the upper spectral radius the greatest lower bound of the set. Finally, we demonstrate the full power of these Krein-Rutman theorems on some PDE examples such as elliptic eigenvalue problems on non-smooth domains, and cooperative systems which may or may not be fully coupled, by using as few PDE tools as possible.

On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature
Huyuan Chen, Dong Ye and Feng Zhou
2020, 40(6): 3201-3214 doi: 10.3934/dcds.2020125 +[Abstract](1595) +[HTML](478) +[PDF](365.89KB)

The purpose of this paper is to study the solutions of

\begin{document}$ \Delta u +K(x) e^{2u} = 0 \quad{\rm in}\;\; \mathbb{R}^2 $\end{document}

with \begin{document}$ K\le 0 $\end{document}. We introduce the following quantities:

\begin{document}$ \alpha_p(K) = \sup\left\{\alpha \in \mathbb{R}:\, \int_{ \mathbb{R}^2} |K(x)|^p(1+|x|)^{2\alpha p+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1. $\end{document}

Under the assumption \begin{document}$ ({\mathbb H}_1) $\end{document}: \begin{document}$ \alpha_p(K)> -\infty $\end{document} for some \begin{document}$ p>1 $\end{document} and \begin{document}$ \alpha_1(K) > 0 $\end{document}, we show that for any \begin{document}$ 0 < \alpha < \alpha_1(K) $\end{document}, there is a unique solution \begin{document}$ u_\alpha $\end{document} with \begin{document}$ u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}} \big) $\end{document} at infinity and \begin{document}$ \beta\in (0, \, \alpha_1(K)-\alpha) $\end{document}. Furthermore, we show an example \begin{document}$ K_0 \leq 0 $\end{document} such that \begin{document}$ \alpha_p(K_0) = -\infty $\end{document} for any \begin{document}$ p>1 $\end{document} and \begin{document}$ \alpha_1(K_0) > 0 $\end{document}, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution \begin{document}$ u_{\alpha_*} $\end{document} such that \begin{document}$ u_{\alpha_*} -\alpha_*\ln|x| = O(1) $\end{document} at infinity for some \begin{document}$ \alpha_* > 0 $\end{document}, which does not converge to a constant at infinity. This example exhibits a new phenomenon of solution with logarithmic growth, finite total curvature, and non-uniform asymptotic behavior at infinity.

Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $
Guoyuan Chen, Yong Liu and Juncheng Wei
2020, 40(6): 3215-3233 doi: 10.3934/dcds.2019228 +[Abstract](2333) +[HTML](705) +[PDF](387.83KB)

We prove that all harmonic maps from \begin{document}$ {{\mathbb{R}}^{2}} $\end{document} to \begin{document}$ {{\mathbb{S}}^{2}} $\end{document} with finite energy are nondegenerate. That is, for any harmonic map \begin{document}$ u $\end{document} from \begin{document}$ {{\mathbb{R}}^{2}} $\end{document} to \begin{document}$ {{\mathbb{S}}^{2}} $\end{document} of degree \begin{document}$ m\in\mathbb Z $\end{document}, all bounded kernel maps of the linearized operator \begin{document}$ L_u $\end{document} at \begin{document}$ u $\end{document} are generated by those harmonic maps near \begin{document}$ u $\end{document} and hence the real dimension of bounded kernel space of \begin{document}$ L_u $\end{document} is \begin{document}$ 4|m|+2 $\end{document}.

A Hopf lemma and regularity for fractional $ p $-Laplacians
Wenxiong Chen, Congming Li and Shijie Qi
2020, 40(6): 3235-3252 doi: 10.3934/dcds.2020034 +[Abstract](2522) +[HTML](535) +[PDF](403.05KB)

In this paper, we study qualitative properties of the fractional \begin{document}$ p $\end{document}-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional \begin{document}$ p- $\end{document}Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure \begin{document}$ (-\triangle)_p^s u\in C^1(\mathbb{R}^n) $\end{document} for smooth functions \begin{document}$ u $\end{document}.

Boundary spike of the singular limit of an energy minimizing problem
Xinfu Chen, Huiqiang Jiang and Guoqing Liu
2020, 40(6): 3253-3290 doi: 10.3934/dcds.2020124 +[Abstract](1589) +[HTML](467) +[PDF](461.47KB)

In this paper, we consider the singular limit of an energy minimizing problem which is a semi-limit of a singular elliptic equation modeling steady states of thin film equation with both Van der Waals force and Born repulsion force. We show that the singular limit of energy minimizers is a Dirac mass located on the boundary point with the maximum curvature.

On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen and Chih-Her Chen
2020, 40(6): 3291-3304 doi: 10.3934/dcds.2020127 +[Abstract](1645) +[HTML](479) +[PDF](382.66KB)

In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [18]), we prove the existence of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory.

Sectional symmetry of solutions of elliptic systems in cylindrical domains
Lucio Damascelli and Filomena Pacella
2020, 40(6): 3305-3325 doi: 10.3934/dcds.2020045 +[Abstract](1661) +[HTML](458) +[PDF](418.61KB)

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in [6,9,16,18].

Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $
Manuel del Pino, Monica Musso, Juncheng Wei and Yifu Zhou
2020, 40(6): 3327-3355 doi: 10.3934/dcds.2020052 +[Abstract](2323) +[HTML](526) +[PDF](494.08KB)

We consider the Cauchy problem for the energy critical heat equation

\begin{document}$ \begin{equation} \left\{ \begin{aligned} u_t & = \Delta u + u^3 {\quad\hbox{in } }\ \mathbb R^4 \times (0, T), \\ u(\cdot, 0) & = u_0 {\quad\hbox{in } } \mathbb R^4. \end{aligned}\right. ~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation} $\end{document}

We find that for given points \begin{document}$ q_1, q_2, \ldots, q_k $\end{document} and any sufficiently small \begin{document}$ T>0 $\end{document} there is an initial condition \begin{document}$ u_0 $\end{document} such that the solution \begin{document}$ u(x, t) $\end{document} of (1) blows up at exactly those \begin{document}$ k $\end{document} points with a type Ⅱ rate, namely larger than \begin{document}$ (T-t)^{-\frac 12} $\end{document}. In fact \begin{document}$ \|u(\cdot, t)\|_\infty \sim (T-t)^{-1}\log^2(T-t) $\end{document}. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.

Spike solutions for a mass conservation reaction-diffusion system
Shin-Ichiro Ei and Shyuh-Yaur Tzeng
2020, 40(6): 3357-3374 doi: 10.3934/dcds.2020049 +[Abstract](1958) +[HTML](504) +[PDF](372.45KB)

This article deals with a mass conservation reaction-diffusion system. As a model for studying cell polarity, we are interested in the existence of spike solutions and some properties related to its dynamics. Variational arguments will be employed to investigate the existence questions. The profile of a spike solution looks like a standing pulse. In addition, the motion of such spikes in heterogeneous media will be derived.

Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity
Maho Endo, Yuki Kaneko and Yoshio Yamada
2020, 40(6): 3375-3394 doi: 10.3934/dcds.2020033 +[Abstract](2653) +[HTML](550) +[PDF](208.91KB)

This paper deals with a free boundary problem for a reaction-diffusion equation in a one-dimensional interval whose boundary consists of a fixed end-point and a moving one. We put homogeneous Dirichlet condition at the fixed boundary, while we assume that the dynamics of the moving boundary is governed by the Stefan condition. Such free boundary problems have been studied by a lot of researchers. We will take a nonlinear reaction term of positive bistable type which exhibits interesting properties of solutions such as multiple spreading phenomena. In fact, it will be proved that large-time behaviors of solutions can be classified into three types; vanishing, small spreading and big spreading. Some sufficient conditions for these behaviors are also shown. Moreover, for two types of spreading, we will give sharp estimates of spreading speed of each free boundary and asymptotic profiles of each solution.

Large time behavior of ODE type solutions to nonlinear diffusion equations
Junyong Eom and Kazuhiro Ishige
2020, 40(6): 3395-3409 doi: 10.3934/dcds.2019229 +[Abstract](2687) +[HTML](697) +[PDF](363.45KB)

Consider the Cauchy problem for a nonlinear diffusion equation

\begin{document}$ \begin{equation} \left\{ \begin{array}{ll} \partial_t u = \Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0) = \lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} $\end{document}

where \begin{document}$ m>0 $\end{document}, \begin{document}$ \alpha\in(-\infty,1) $\end{document}, \begin{document}$ \lambda>0 $\end{document} and \begin{document}$ \varphi\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N) $\end{document} with \begin{document}$ 1\le r<\infty $\end{document} and \begin{document}$ \inf_{x\in{\bf R}^N}\varphi(x)>-\lambda $\end{document}. Then the positive solution to problem (P) behaves like a positive solution to ODE \begin{document}$ \zeta' = \zeta^\alpha $\end{document} in \begin{document}$ (0,\infty) $\end{document} and it tends to \begin{document}$ +\infty $\end{document} as \begin{document}$ t\to\infty $\end{document}. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.

Asymptotic population abundance of a two-patch system with asymmetric diffusion
Mengting Fang, Yuanshi Wang, Mingshu Chen and Donald L. DeAngelis
2020, 40(6): 3411-3425 doi: 10.3934/dcds.2020031 +[Abstract](1808) +[HTML](547) +[PDF](622.93KB)

This paper considers a two-patch system with asymmetric diffusion rates, in which exploitable resources are included. By using dynamical system theory, we exclude periodic solution in the one-patch subsystem and demonstrate its global dynamics. Then we exhibit uniform persistence of the two-patch system and demonstrate uniqueness of the positive equilibrium, which is shown to be asymptotically stable when the diffusion rates are sufficiently large. By a thorough analysis on the asymptotic population abundance, we demonstrate necessary and sufficient conditions under which the asymmetric diffusion rates can lead to the result that total equilibrium population abundance in heterogeneous environments is larger than that in heterogeneous/homogeneous environments with no diffusion, which is not intuitive. Our result extends previous work to the situation of asymmetric diffusion and provides new insights. Numerical simulations confirm and extend our results.

On the viscous Camassa-Holm equations with fractional diffusion
Zaihui Gan, Fanghua Lin and Jiajun Tong
2020, 40(6): 3427-3450 doi: 10.3934/dcds.2020029 +[Abstract](2133) +[HTML](478) +[PDF](414.4KB)

We study Cauchy problem of a class of viscous Camassa-Holm equations (or Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and in the whole space in two and three dimensions. Order of the fractional diffusion is assumed to be \begin{document}$ 2s $\end{document} with \begin{document}$ s\in [n/4,1) $\end{document}, which seems to be sharp for the validity of the main results of the paper; here \begin{document}$ n = 2,3 $\end{document} is the dimension of space. We prove global well-posedness in \begin{document}$ C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2})) $\end{document} whenever the initial data \begin{document}$ u_0\in D(A) $\end{document}, where \begin{document}$ A $\end{document} is the Stokes operator. We also prove that such global solutions gain regularity instantaneously after the initial time. A bound on a higher-order spatial norm is also obtained.

The sign of traveling wave speed in bistable dynamics
Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara and Chang-Hong Wu
2020, 40(6): 3451-3466 doi: 10.3934/dcds.2020047 +[Abstract](2355) +[HTML](533) +[PDF](332.61KB)

We are concerned with the sign of traveling wave speed in bistable dynamics. This question is related to which species wins the competition in multiple species competition models. It is well-known that the wave speed is unique for traveling wave connecting two stable states. In this paper, we first review some known results on the sign of wave speed in bistable two species competition models. Then we derive rigorously the sign of bistable wave speed for a special three species competition model describing the competition in two different circumstances: (1) two species are weak competitors and one species is a strong competitor; (2) three species are very strong competitors. It is interesting to observe that, under certain conditions on the parameters, two weaker competitors can wipe out the strongest competitor.

A stage structured model of delay differential equations for Aedes mosquito population suppression
Mugen Huang, Moxun Tang, Jianshe Yu and Bo Zheng
2020, 40(6): 3467-3484 doi: 10.3934/dcds.2020042 +[Abstract](2732) +[HTML](751) +[PDF](515.07KB)

Tremendous efforts have been devoted to the development and analysis of mathematical models to assess the efficacy of the endosymbiotic bacterium Wolbachia in the control of infectious diseases such as dengue and Zika, and their transmission vector Aedes mosquitoes. However, the larval stage has not been included in most models, which causes an inconvenience in testing directly the density restriction on population growth. In this work, we introduce a system of delay differential equations, including both the adult and larval stages of wild mosquitoes, interfered by Wolbachia infected males that can cause complete female sterility. We clarify its global dynamics rather completely by using delicate analyses, including a construction of Liapunov-type functions, and determine the threshold level \begin{document}$ R_0 $\end{document} of infected male releasing. The wild population is suppressed completely if the releasing level exceeds \begin{document}$ R_0 $\end{document} uniformly. The dynamical complexity revealed by our analysis, such as bi-stability and semi-stability, is further exhibited through numerical examples. Our model generates a temporal profile that captures several critical features of Aedes albopictus population in Guangzhou from 2011 to 2016. Our estimate for optimal mosquito control suggests that the most cost-efficient releasing should be started no less than 7 weeks before the peak dengue season.

Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms
Yunfeng Jia, Yi Li, Jianhua Wu and Hong-Kun Xu
2020, 40(6): 3485-3507 doi: 10.3934/dcds.2019227 +[Abstract](2504) +[HTML](752) +[PDF](416.37KB)

We consider the structure and the stability of positive radial solutions of a semilinear inhomogeneous elliptic equation with multiple growth terms

\begin{document}$ \Delta u+\sum\limits_{i = 1}^{k}K_i(|x|)u^{p_i}+\mu f(|x|) = 0, \quad x\in\mathbb{R}^n, $\end{document}

which is a generalization of Matukuma's equation describing the dynamics of a globular cluster of stars. Equations similar to this kind have come up both in geometry and in physics, and have been a subject of extensive studies. Our result shows that any positive radial solution is stable or weakly asymptotically stable with respect to certain norm.

Global stabilization of the full attraction-repulsion Keller-Segel system
Hai-Yang Jin and Zhi-An Wang
2020, 40(6): 3509-3527 doi: 10.3934/dcds.2020027 +[Abstract](2333) +[HTML](518) +[PDF](396.76KB)

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system

\begin{document}$\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \nabla \cdot (\chi u\nabla v) + \nabla \cdot (\xi u\nabla w),}&{x \in \Omega ,t > 0,}\\{{v_t} = {D_1}\Delta v + \alpha u - \beta v,}&{x \in \Omega ,t > 0,}\\{{w_t} = {D_2}\Delta w + \gamma u - \delta w,}&{x \in \Omega ,t > 0,}\\{u(x,0) = {u_0}(x),v(x,0) = {v_0}(x),w(x,0) = {w_0}(x)}&{x \in \Omega ,}\end{array}} \right.\;\;\;\;\left( * \right)$\end{document}

in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document} with smooth boundary subject to homogeneous Neumann boundary conditions. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system (*) with large initial data \begin{document}$ (u_0,v_0,w_0) \in [W^{1,\infty}(\Omega)]^3 $\end{document}. Precisely, we show that if the parameters satisfy \begin{document}$ \frac{\xi\gamma}{\chi\alpha}\geq \max\Big\{\frac{D_1}{D_2},\frac{D_2}{D_1},\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $\end{document} for all positive parameters \begin{document}$ D_1,D_2,\chi,\xi,\alpha,\beta,\gamma $\end{document} and \begin{document}$ \delta $\end{document}, the system (*) has a unique global classical solution \begin{document}$ (u,v,w) $\end{document}, which converges to the constant steady state \begin{document}$ (\bar{u}_0,\frac{\alpha}{\beta}\bar{u}_0,\frac{\gamma}{\delta}\bar{u}_0) $\end{document} as \begin{document}$ t\to+\infty $\end{document}, where \begin{document}$ \bar{u}_0 = \frac{1}{|\Omega|}\int_\Omega u_0dx $\end{document}. Furthermore, the decay rate is exponential if \begin{document}$ \frac{\xi\gamma}{\chi\alpha}> \max\Big\{\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $\end{document}. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. \begin{document}$ D_1\ne D_2 $\end{document}) in multi-dimensions.

Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere
Yoshitsugu Kabeya
2020, 40(6): 3529-3559 doi: 10.3934/dcds.2020040 +[Abstract](1798) +[HTML](578) +[PDF](387.07KB)

We consider the eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a spherical domain. Especially, we investigate the case when the domain is a large zonal one and letting the zone larger so that the zone covers the whole sphere as a limit. We discuss the behavior of eigenvalues according to the rate of expansion of the zone.

On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates
Yukio Kan-On
2020, 40(6): 3561-3570 doi: 10.3934/dcds.2020161 +[Abstract](1322) +[HTML](444) +[PDF](1908.79KB)

In 1979, Shigesada, Kawasaki and Teramoto [11] proposed a mathematical model with nonlinear diffusion, to study the segregation phenomenon in a two competing species community. In this paper, we discuss limiting systems of the model as the cross-diffusion rates included in the nonlinear diffusion tend to infinity. By formal calculation without rigorous proof, we obtain one limiting system which is a little different from that established in Lou and Ni [5].

Asymmetric dispersal and evolutional selection in two-patch system
Yong-Jung Kim, Hyowon Seo and Changwook Yoon
2020, 40(6): 3571-3593 doi: 10.3934/dcds.2020043 +[Abstract](2164) +[HTML](598) +[PDF](745.65KB)

Biological organisms leave their habitat when the environment becomes harsh. The essence of a biological dispersal is not in the rate, but in the capability to adjust to the environmental changes. In nature, conditional asymmetric dispersal strategies appear due to the spatial and temporal heterogeneity in the environment. Authors show that such a dispersal strategy is evolutionary selected in the context two-patch problem of Lotka-Volterra competition model. They conclude that, if a conditional asymmetric dispersal strategy is taken, the dispersal is not necessarily disadvantageous even for the case that there is no temporal fluctuation of environment at all.

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems
Alexandra Köthe, Anna Marciniak-Czochra and Izumi Takagi
2020, 40(6): 3595-3627 doi: 10.3934/dcds.2020170 +[Abstract](1866) +[HTML](813) +[PDF](1079.38KB)

The paper is devoted to analysis of far-from-equilibrium pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.

Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation
Joel Kübler and Tobias Weth
2020, 40(6): 3629-3656 doi: 10.3934/dcds.2020032 +[Abstract](2033) +[HTML](496) +[PDF](484.63KB)

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem

\begin{document}$ \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -\Delta u & = |x|^\alpha |u|^{p-2}u&&\qquad \text{in $ {\bf B}$,}\\ u& = 0&&\qquad \text{on $\partial {\bf B}$,} \end{aligned} \right. \end{equation*} $\end{document}

in the unit ball \begin{document}$ {\bf B} \subset \mathbb{R}^N,N\geq 3 $\end{document}, \begin{document}$ p>2 $\end{document} in the limit \begin{document}$ \alpha \to +\infty $\end{document}. More precisely, for a given positive integer \begin{document}$ K $\end{document}, we derive asymptotic \begin{document}$ C^1 $\end{document}-expansions for the negative eigenvalues of the linearization of the unique radial solution \begin{document}$ u_\alpha $\end{document} of \begin{document}$ (H) $\end{document} with precisely \begin{document}$ K $\end{document} nodal domains and \begin{document}$ u_\alpha(0)>0 $\end{document}. As an application, we derive the existence of an unbounded sequence of bifurcation points on the radial solution branch \begin{document}$ \alpha \mapsto (\alpha,u_\alpha) $\end{document} which all give rise to bifurcation of nonradial solutions whose nodal sets remain homeomorphic to a disjoint union of concentric spheres.

Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion
Qing Li and Yaping Wu
2020, 40(6): 3657-3682 doi: 10.3934/dcds.2020051 +[Abstract](1866) +[HTML](459) +[PDF](693.91KB)

This paper is concerned with the existence and stability of nontrivial positive steady states of Shigesada-Kawasaki-Teramoto competition model with cross diffusion under zero Neumann boundary condition. By applying the special perturbation argument based on the Lyapunov-Schmidt reduction method, we obtain the existence and the detailed asymptotic behavior of two branches of nontrivial large positive steady states for the specific shadow system when the random diffusion rate of one species is near some critical value. Further by applying the detailed spectral analysis with the special perturbation argument, we prove the spectral instability of the two local branches of nontrivial positive steady states for the limiting system. Finally, we prove the existence and instability of the two branches of nontrivial positive steady states for the original SKT cross-diffusion system when both the cross diffusion rate and random diffusion rate of one species are large enough, while the random diffusion rate of another species is near some critical value.

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach
Qian Liu, Shuang Liu and King-Yeung Lam
2020, 40(6): 3683-3714 doi: 10.3934/dcds.2020050 +[Abstract](2046) +[HTML](464) +[PDF](766.32KB)

We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.

A functional approach towards eigenvalue problems associated with incompressible flow
Shuang Liu and Yuan Lou
2020, 40(6): 3715-3736 doi: 10.3934/dcds.2020028 +[Abstract](1920) +[HTML](542) +[PDF](453.64KB)

We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator \begin{document}$ L_{A} = -\mathrm{div}(a(x)\nabla )+A\mathbf{V}\cdot\nabla +c(x) $\end{document} and its adjoint operator for general incompressible flow \begin{document}$ \mathbf{V} $\end{document}. The functional can be applied to establish the monotonicity of the principal eigenvalue \begin{document}$ \lambda_1(A) $\end{document}, as a function of the advection amplitude \begin{document}$ A $\end{document}, for the operator \begin{document}$ L_{A} $\end{document} subject to Dirichlet, Robin and Neumann boundary conditions. This gives a new proof of a conjecture raised by Berestycki, Hamel and Nadirashvili [5]. The functional can also be used to prove the monotonicity of the normalized speed \begin{document}$ c^{*}(A)/A $\end{document} for general incompressible flow, where \begin{document}$ c^{*}(A) $\end{document} is the minimal speed of traveling fronts. This extends an earlier result of Berestycki [3] for steady shear flow.

Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature
Tianwen Luo, Tao Tao and Liqun Zhang
2020, 40(6): 3737-3765 doi: 10.3934/dcds.2019230 +[Abstract](2554) +[HTML](711) +[PDF](448.81KB)

We show the existence of finite kinetic energy solution with prescribed kinetic energy to the 2d Boussinesq equations with diffusive temperature on torus.

Perturbative techniques for the construction of spike-layers
Andrea Malchiodi
2020, 40(6): 3767-3787 doi: 10.3934/dcds.2020055 +[Abstract](1709) +[HTML](537) +[PDF](390.03KB)

In this paper we survey some results concerning the construction of spike-layers, namely solutions to singularly perturbed equations that exhibit a concentration behaviour. Their study is motivated by the analysis of pattern formation in biological systems such as the Keller-Segel or the Gierer-Meinhardt's. We describe some general perturbative variational strategy useful to study concentration at points, and also at spheres in radially symmetric situations.

Asymptotic homogenization for delay-differential equations and a question of analyticity
John Mallet-Paret and Roger D. Nussbaum
2020, 40(6): 3789-3812 doi: 10.3934/dcds.2020044 +[Abstract](1726) +[HTML](556) +[PDF](402.96KB)

We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches \begin{document}$ +\infty $\end{document} as \begin{document}$ t\to\pm\infty $\end{document}. Typical simple examples are

\begin{document}$ x'(t) = \sin (t^q)x(t-1) \qquad\text{and}\qquad x'(t) = e^{it^q}x(t-1), \;\;\;\;\;\;\;\;{(*)} $\end{document}

where \begin{document}$ q\ge 2 $\end{document} is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit \begin{document}$ \lim\limits_{t\to-\infty}x(t) = x_- $\end{document} at \begin{document}$ -\infty $\end{document}. We also show, under appropriate conditions, that any solution of an initial value problem has a finite limit \begin{document}$ \lim\limits_{t\to+\infty}x(t) = x_+ $\end{document} at \begin{document}$ +\infty $\end{document}, and thus we establish the existence of a class of heteroclinic solutions. We term this limiting phenomenon, and thus the existence of such solutions, "asymptotic homogenization." Note that in general, proving the existence of a bounded solution of a given delay-differential equation on a semi-infinite interval \begin{document}$ (-\infty,-T] $\end{document} is often highly nontrivial.

Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution \begin{document}$ x: \mathbb{R}\to \mathbb{C} $\end{document} of one of the equations in \begin{document}$ (*) $\end{document} with limits \begin{document}$ x_\pm $\end{document} at \begin{document}$ \pm\infty $\end{document} is \begin{document}$ C^\infty $\end{document}, but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in \begin{document}$ (*) $\end{document} can be extended to the lower half plane \begin{document}$ \{z\in \mathbb{C}\:|\: \mathop{{{\rm{Im}}}} z<0\} $\end{document} as an analytic function.

Turing type instability in a diffusion model with mass transport on the boundary
Yoshihisa Morita and Kunimochi Sakamoto
2020, 40(6): 3813-3836 doi: 10.3934/dcds.2020160 +[Abstract](1927) +[HTML](484) +[PDF](355.44KB)

Some reaction-diffusion models describing the cell polarity are proposed, where the system has two independent variables standing for the concentration of proteins in the membrane and the cytosol respectively. In this article we deal with such a polarity model consisting of one equation on a unit sphere and the other one in the ball inside the sphere. The two equations are coupled through a nonlinear boundary condition and the total mass is conserved. We investigate the linearized stability of a constant steady state and provide conditions under which a Turing type instability takes place, namely, the constant state is stable against spatially uniform perturbations on the sphere for all choices of diffusion rates, while unstable against nonuniform perturbations on the sphere as the diffusion coefficient of the equation on the sphere becomes small relative to the one in the ball.

Indefinite nonlinear diffusion problem in population genetics
Kimie Nakashima
2020, 40(6): 3837-3855 doi: 10.3934/dcds.2020169 +[Abstract](1608) +[HTML](411) +[PDF](374.36KB)

We study the following Neumann problem in one dimension,

\begin{document}$ \left\{ {\begin{array}{*{20}{l}}\begin{array}{l}{u_t} = du'' + g(x){u^2}(1 - u)\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\0 \le u \le 1\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\u'(0,t) = u'(1,t) = 0\quad {\rm{in}}\quad (0,\infty ),\end{array}\end{array}} \right.$\end{document}

where \begin{document}$ g $\end{document} changes sign in \begin{document}$ (0, 1) $\end{document}. This equation models the "complete dominance" case in population genetics of two alleles. It is known that this equation has a nontrivial stable steady state \begin{document}$ U_d $\end{document} for \begin{document}$ d $\end{document} sufficiently small. We show that \begin{document}$ U_d $\end{document} is a unique nontrivial steady state under a condition \begin{document}$ \int_{0}^1\, g(x)\, dx\geq 0 $\end{document} and some other additional condition.

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth
Gabrielle Nornberg, Delia Schiera and Boyan Sirakov
2020, 40(6): 3857-3881 doi: 10.3934/dcds.2020128 +[Abstract](2039) +[HTML](437) +[PDF](504.48KB)

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as

\begin{document}$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle = \lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x), $\end{document}

for \begin{document}$ i = 1, \cdots, n $\end{document}, in a bounded \begin{document}$ C^{1, 1} $\end{document} domain \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} with Dirichlet boundary conditions; here \begin{document}$ n\geq 1 $\end{document}, \begin{document}$ \lambda \in \mathbb{R} $\end{document}, \begin{document}$ c_{ij}, \, h_i \in L^\infty(\Omega) $\end{document}, \begin{document}$ c_{ij}\geq 0 $\end{document}, \begin{document}$ M_i $\end{document} satisfies \begin{document}$ 0<\mu_1 I\leq M_i\leq \mu_2 I $\end{document}, and \begin{document}$ F_i $\end{document} is an uniformly elliptic Isaacs operator.

We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.

Convergence and structure theorems for order-preserving dynamical systems with mass conservation
Toshiko Ogiwara, Danielle Hilhorst and Hiroshi Matano
2020, 40(6): 3883-3907 doi: 10.3934/dcds.2020129 +[Abstract](1876) +[HTML](607) +[PDF](407.73KB)

We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which \begin{document}$ 0 $\end{document} is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczyński (1987) and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.

Variational and operator methods for Maxwell-Stokes system
Xing-Bin Pan
2020, 40(6): 3909-3955 doi: 10.3934/dcds.2020036 +[Abstract](2132) +[HTML](761) +[PDF](545.81KB)

In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.

The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs
Xiaofeng Ren and David Shoup
2020, 40(6): 3957-3979 doi: 10.3934/dcds.2020048 +[Abstract](1576) +[HTML](492) +[PDF](603.83KB)

When the Ohta-Kawasaki theory for diblock copolymers is applied to a bounded domain with the Neumann boundary condition, one faces the possibility of micro-domain interfaces intersecting the system boundary. In a particular parameter range, there exist stationary assemblies, stable in some sense, that consist of both perturbed discs in the interior of the system and perturbed half discs attached to the boundary of the system. The circular arcs of the half discs meet the system boundary perpendicularly. The number of the interior discs and the number of the boundary half discs are arbitrarily prescribed and their radii are asymptotically the same. The locations of these discs and half discs are determined by the minimization of a function related to the Green's function of the Laplace operator with the Neumann boundary condition. Numerical calculations based on the theoretical findings show that boundary half discs help lower the energy of stationary assemblies.

Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations
Masaharu Taniguchi
2020, 40(6): 3981-3995 doi: 10.3934/dcds.2020126 +[Abstract](1730) +[HTML](431) +[PDF](419.08KB)

For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre [4]. This paper gives another proof of the existence of axisymmetric traveling fronts. Our method is as follows. We use pyramidal traveling fronts for unbalanced reaction-diffusion equations, and take the balanced limit. Then we obtain axisymmetric traveling fronts in a balanced bistable reaction-diffusion equation. Since pyramidal traveling fronts have been studied in many equations or systems, our method might be applicable to study axisymmetric traveling fronts in these equations or systems.

On space-time periodic solutions of the one-dimensional heat equation
Dong-Ho Tsai and Chia-Hsing Nien
2020, 40(6): 3997-4017 doi: 10.3934/dcds.2020037 +[Abstract](1842) +[HTML](519) +[PDF](444.59KB)

We look for solutions \begin{document}$ u\left( x,t\right) $\end{document} of the one-dimensional heat equation \begin{document}$ u_{t} = u_{xx} $\end{document} which are space-time periodic, i.e. they satisfy the property

\begin{document}$ u\left( x+a,t+b\right) = u\left( x,t\right) $\end{document}

for all \begin{document}$ \left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), $\end{document} and derive their Fourier series expansions. Here \begin{document}$ a\geq0,\ b\geq 0 $\end{document} are two constants with \begin{document}$ a^{2}+b^{2}>0. $\end{document} For general equation of the form \begin{document}$ u_{t} = u_{xx}+Au_{x}+Bu, $\end{document} where \begin{document}$ A,\ B $\end{document} are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when \begin{document}$ B>0 $\end{document} and is given by a linear combination of \begin{document}$ \cos\left( \sqrt{B}\left( x+At\right) \right) $\end{document} and \begin{document}$ \sin\left( \sqrt{B}\left( x+At\right) \right). $\end{document}

Monotone and nonmonotone clines with partial panmixia across a geographical barrier
Yantao Wang and Linlin Su
2020, 40(6): 4019-4037 doi: 10.3934/dcds.2020056 +[Abstract](1660) +[HTML](562) +[PDF](902.38KB)

The number of clines (i.e., nonconstant equilibria) maintained by viability selection, migration, and partial global panmixia in a step-environment with a geographical barrier is investigated. Our results extend the results of T. Nagylaki (2016, Clines with partial panmixia across a geographical barrier, Theor. Popul. Biol. 109) from the no dominance case to arbitrary dominance and to various other selection functions. Unexpectedly, besides the usual monotone clines, we discover nonmonotone clines with both equal and unequal limits at \begin{document}$ \pm\infty $\end{document}.

Refined regularity and stabilization properties in a degenerate haptotaxis system
Michael Winkler and Christian Stinner
2020, 40(6): 4039-4058 doi: 10.3934/dcds.2020030 +[Abstract](1371) +[HTML](461) +[PDF](401.32KB)

We consider the degenerate haptotaxis system

\begin{document}$ \left\{ \begin{array}{l} u_t = (d(x)u)_{xx} - (d(x)uw_x )_x, \\ w_t = -ug(w), \end{array} \right. $\end{document}

endowed with no-flux boundary conditions in a bounded open interval \begin{document}$ \Omega \subset \mathbb{R} $\end{document}. It was proposed as a basic model for haptotactic migration in heterogeneous environments. If the diffusion is degenerate in the sense that \begin{document}$ d $\end{document} is non-negative, has a non-empty zero set and satisfies \begin{document}$ \int_\Omega \frac{1}{d} <\infty $\end{document}, then it has been shown in [12] under appropriate assumptions on the initial data that the system has a global generalized solution satisfying in particular \begin{document}$ u(\cdot,t) \rightharpoonup \frac{\mu_\infty}{d} $\end{document} weakly in \begin{document}$ L^1 (\Omega) $\end{document} as \begin{document}$ t \to \infty $\end{document} for some positive constant \begin{document}$ \mu_\infty $\end{document}.

We now prove that under the additional restriction \begin{document}$ \int_\Omega \frac{1}{d^2} <\infty $\end{document} we have the strong convergence \begin{document}$ u(\cdot,t)\to \frac{\mu_\infty}{d} $\end{document} in \begin{document}$ L^p (\Omega) $\end{document} as \begin{document}$ t \to \infty $\end{document} for any \begin{document}$ p \in (1,2) $\end{document}. In addition, with the same restriction on \begin{document}$ d $\end{document} we obtain improved regularity properties of \begin{document}$ u $\end{document}, for instance \begin{document}$ du \in L^\infty ((0,\infty); L^p(\Omega)) $\end{document} for any \begin{document}$ p \in (1,\infty) $\end{document}.

Simulation of post-hurricane impact on invasive species with biological control management
Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis and Bo Zhang
2020, 40(6): 4059-4071 doi: 10.3934/dcds.2020038 +[Abstract](1947) +[HTML](636) +[PDF](842.56KB)

Understanding the effects of hurricanes and other large storms on ecological communities and the post-event recovery in these communities can guide management and ecosystem restoration. This is particularly important for communities impacted by invasive species, as the hurricane may affect control efforts. Here we consider the effect of a hurricane on tree communities in southern Florida that has been invaded by Melaleuca quinquevervia (melaleuca), an invasive Australian tree. Biological control agents were introduced starting in the 1990s and are reducing melaleuca in habitats where they are established. We used size-structured matrix modeling as a tool to project the continued possible additional effects of a hurricane on a pure stand of melaleuca that already had some level of biological control. The model results indicate that biological control could suppress or eliminate melaleuca within decades. A hurricane that does severe damage to the stand may accelerate the trend toward elimination of melaleuca with both strong and moderate biological control. However, if the biological control is weak, the stand is resilient to all but extremely severe hurricane damage. Although only a pure melaleuca stand was simulated in this study, other plants, such as natives, are likely to accelerate the decline of melaleuca due to competition. Our model provides a new tool to simulate post-hurricanes effect on invasive species and highlights the essential role that biological control has played on invasive species management.

2021 Impact Factor: 1.588
5 Year Impact Factor: 1.568
2021 CiteScore: 2.4




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