A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch

We will analyze the symmetric positive solutions to the two-point steady state reaction-diffusion equation: \begin{equation*} \begin{split} -&u^{\prime \prime}= \begin{cases} \lambda\left[ u-\dfrac{1}{K}u^2-\dfrac{cu^2}{1+u^2}\right];& x\in [L,1-L] ,\\ \lambda \left[u-\dfrac{1}{K}u^2\right];& x\in (0,L)\cup(1-L,1), \end{cases} \\ -&u^{\prime}(0) + \sqrt{\lambda}\gamma u(0) = 0,\\ &u^{\prime}(1) + \sqrt{\lambda}\gamma u(1) = 0,\\ \end{split} \end{equation*} where $\lambda$, $c$, $K$, and $\gamma$ are positive parameters and the parameter $L\in(0,\frac{1}{2})$. The steady state reaction-diffusion equation above occurs in ecological systems and population dynamics. The above model exhibits logistic growth in the one-dimensional habitat $\Omega_0=(0,1)$, where grazing (type of predation) is occurring on the subregion $[L,1-L]$. In this model, $u$ is the population density and $c$ is the maximum grazing rate. $\lambda$ is a parameter which influences the equation as well as the boundary conditions, and $\gamma$ represents the hostility factor of the surrounding matrix. Previous studies have shown the occurrence of S-shaped bifurcation curves for positive solutions for certain parameter ranges when the boundary condition is Dirichlet ($\gamma \longrightarrow \infty$). Here we discuss the occurrence of S-shaped bifurcation curves for certain parameter ranges, when $\gamma$ is finite, and their evolutions as $\gamma$ and $L$ vary

A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch

We will analyze the symmetric positive solutions to the two-point steady state reaction-diffusion equation: −u 00 = u − 1 K u 2 − cu2 1 + u 2 ; x ∈ [L, 1 − L], u − 1 K u 2 ; x ∈ (0, L) ∪ (1 − L, 1), −u 0 (0) + √ λγu(0) = 0, u 0 (1) + √ λγu(1) = 0, where λ, c, K, and γ are positive parameters and the parameter L ∈ (0, 1 2 ). The steady state reaction-diffusion equation above occurs in ecological systems and population dynamics. The above model exhibits logistic growth in the one-dimensional habitat Ω0 = (0, 1), where grazing (type of predation) is occurring on the subregion [L, 1 − L]. In this model, u is the population density and c is the maximum grazing rate. λ is a parameter which influences the equation as well as the boundary conditions, and γ represents the hostility factor of the surrounding matrix. Previous studies have shown the occurrence of S-shaped bifurcation curves for positive solutions for certain parameter ranges when the boundary condition is Dirichlet (γ −→ ∞). Here we discuss the occurrence of S-shaped bifurcation curves for certain parameter ranges, when γ is finite, and their evolutions as γ and L vary

Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0) The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect. In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density. The habitat patch is one-dimensional and is surrounded by a tuneable hostile matrix. We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies. Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them. Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch. Our results are obtained mathematically through the method of sub-super solutions, time map analysis, and numerical computations using Wolfram Mathematica

Mathematical and computational analysis of reaction diffusion models arising in ecology

The focus of this thesis is to study long term solutions for classes of steady state reaction diffusion equations. In particular, we study reaction diffusion models arising in mathematical ecology. We study how the patch size affects the existence, nonexistence, multiplicity, and uniqueness of the steady states. Our focus is also to study how various forms of density dependent emigrations at the boundary, and the effective matrix hostility, affect steady states. These considerations lead to the study of various forms of nonlinear boundary conditions. Further, they lead to the study of reaction diffusion models where a parameter (related to the patch size) gets involved in the differential equation as well as the boundary conditions. We establish analytical results in any dimension, namely, establish existence, nonexistence, multiplicity, and uniqueness results. Our existence and multiplicity results are achieved by a method of sub-supersolutions and uniqueness results via comparison principles and a-priori estimates. Via computational methods, we also obtain exact bifurcation diagrams describing the structure of the steady states. Namely, we obtain these bifurcation diagrams via a modified quadrature method and Mathematica computations in the one-dimensional case, and via the use of finite element methods and nonlinear solvers in Matlab in the two-dimensional case. This dissertation aims to significantly enrich the mathematical and computational analysis literature on reaction diffusion models arising in ecology

Mathematical and Computational Analysis of Reaction Diffusion Models Arising in Ecology

The focus of this thesis is to study long term solutions for classes of steady state reaction diffusion equations. In particular, we study reaction diffusion models arising in mathematical ecology. We study how the patch size affects the existence, nonexistence, multiplicity, and uniqueness of the steady states. Our focus is also to study how various forms of density dependent emigrations at the boundary, and the effective matrix hostility, affect steady states. These considerations lead to the study of various forms of nonlinear boundary conditions. Further, they lead to the study of reaction diffusion models where a parameter (related to the patch size) gets involved in the differential equation as well as the boundary conditions. We establish analytical results in any dimension, namely, establish existence, nonexistence, multiplicity, and uniqueness results. Our existence and multiplicity results are achieved by a method of sub-supersolutions and uniqueness results via comparison principles and a-priori estimates. Via computational methods, we also obtain exact bifurcation diagrams describing the structure of the steady states. Namely, we obtain these bifurcation diagrams via a modified quadrature method and Mathematica computations in the one-dimensional case, and via the use of finite element methods and nonlinear solvers in Matlab in the two-dimensional case. This dissertation aims to significantly enrich the mathematical and computational analysis literature on reaction diffusion models arising in ecology

SINGULAR REACTION DIFFUSION EQUATIONS WHERE A PARAMETER INFLUENCES THE REACTION TERM AND THE BOUNDARY CONDITIONS

We analyse positive solutions to the steady state reaction diffusion equation: {-u '' - lambda h(t)f(u) in (0,1), -du'(0) + mu(lambda)u(0) = 0, u'(1) + mu(lambda)u(1) = 0, where lambda > 0 is a parameter, d >= 0 is a constant, f is an element of C-2([0, infinity),R) is an increasing function which is sublinear at infinity (lim(s ->infinity) f(s)/s = 0), h is an element of C-1((0, 1], (0, infinity)) is a nonincreasing function with hi := h(1) > 0 and there exist constants d(0) > 0, alpha is an element of [0, 1) such that h(t) = 0. We consider three cases of f, namely, f (0) = 0, f (0) > 0 and f (0) > 1

Singular reaction diffusion equations where a parameter influences the reaction term and the boundary conditions

We analyse positive solutions to the steady state reaction diffusion equation: \begin{equation*} \label{1.11} \begin{cases} -u''=\lambda h(t) f(u) \quad \text{in } (0,1), \\ -du'(0)+\mu(\lambda) u(0)=0,\\ u'(1)+\mu(\lambda) u(1)=0, \end{cases} \end{equation*} where \lambda> 0 is a parameter, $d\geq 0$ is a constant, $f \in C^2([0,\infty),\mathbb{R})$ is an increasing function which is sublinear at infinity $\Big (\lim\limits_{s \rightarrow \infty}{f(s)}/{s}=0\Big)$, $h \in C^1((0,1],(0,\infty))$ is a nonincreasing function with h_1:=h(1)> 0 and there exist constants d_0> 0, $\alpha \in [0,1)$ such that $h(t)\leq {d_0}/{t^\alpha}$ for all $t \in (0,1]$, and $\mu \in C([0,\infty),[0,\infty))$ is an increasing function such that $\mu(0)\geq 0$. We consider three cases of $f$, namely, $f(0)=0$, f(0)> 0 and f(0)< 0. We will discuss existence and multiplicity results via the method of sub-supersolutions. Further, we will establish uniqueness results for $\lambda\approx 0$ and $\lambda\gg 1$