On the existence and multiplicity of positive solutions to classes of steady state reaction diffusion systems with multiple parameters

We study positive solutions to the steady state reaction diffusion systems of the form: \begin{equation} \left\{\begin{array}{ll} -\Delta u = \lambda f(v)+\mu h(u), & \Omega,\\ -\Delta v = \lambda g(u)+\mu q(v),& \Omega,\\ \frac{\partial u}{\partial \eta}+\sqrt[]{\lambda +\mu}\, u=0,& \partial\Omega,\\ \frac{\partial v}{\partial \eta}+\sqrt[]{\lambda +\mu}\, v=0, & \partial\Omega,\\ \end{array}\right. \end{equation} where ${\lambda,\mu>0}$ are positive parameters, ${\Omega}$ is a bounded in $\mathbb{R}^{N}$$(N>1)$ with smooth boundary ${\partial \Omega}$, or ${\Omega=(0,1)}$, ${ \frac{\partial z}{\partial \eta} }$ is the outward normal derivative of $z$. Here $f, g, h, q\in C^{2} [0,r)\cap C[0,\infty)$ for some $r>0$. Further, we assume that $f, g, h$ and $q$ are increasing functions such that $f(0) = g(0) = h(0) = {q}(0) = 0$, $f^\prime(0), g^\prime(0), h^\prime(0), q^\prime(0) > 0$, and $\lim\limits_{s\to \infty}\frac{f(M g(s))}{s}=0$ for all $M>0$. Under certain additional assumptions on $f, g, h$ and $q$ we prove our existence and multiplicity results. Our existence and multiplicity results are proved using sub-super solution methods

Alternate Stable States in Ecological Systems

In this thesis we study two reaction-diffusion models that have been used to analyze the existence of alternate stable states in ecosystems. The first model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. The second model describes phosphorus cycling in stratified lakes. The same equation has also been used to describe the colonization of barren soils in drylands by vegetation. In this study we discuss the existence of multiple positive solutions, leading to the occurrence of S-shaped bifurcation curves. We were able to show that both the models have alternate stable states for certain ranges of parameter values. We also introduce a constant yield harvesting term to the first model and discuss the existence of positive solutions including the occurrence of a Sigma-shaped bifurcation curve in the case of a one-dimensional model. Again we were able to establish that for certain ranges of parameter values the model has alternate stable states. Thus we establish analytically that the above models are capable of describing the phenomena of alternate stable states in ecological systems. We prove our results by the method of sub-super solutions and quadrature method

Alternate Stable States in Ecological Systems

In this thesis we study two reaction-diffusion models that have been used to analyze the existence of alternate stable states in ecosystems. The first model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. The second model describes phosphorus cycling in stratified lakes. The same equation has also been used to describe the colonization of barren soils in drylands by vegetation. In this study we discuss the existence of multiple positive solutions, leading to the occurrence of S-shaped bifurcation curves. We were able to show that both the models have alternate stable states for certain ranges of parameter values. We also introduce a constant yield harvesting term to the first model and discuss the existence of positive solutions including the occurrence of a Sigma-shaped bifurcation curve in the case of a one-dimensional model. Again we were able to establish that for certain ranges of parameter values the model has alternate stable states. Thus we establish analytically that the above models are capable of describing the phenomena of alternate stable states in ecological systems. We prove our results by the method of sub-super solutions and quadrature method

Multiple positive solutions of the one-dimensional p-Laplacian

AbstractIn this work we investigate the existence of positive solutions of the p-Laplacian, using the quadrature method. We prove the existence of multiple solutions of the one-dimensional p-Laplacian for α⩾0, and determine their exact number for α=0

Exact Multiplicity of Sign-Changing Solutions for a Class of Second-Order Dirichlet Boundary Value Problem with Weight Function

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problems u″+a(t)f(u)=0, t∈(0, 1), u(0)=0, and u(1)=0, where f∈C(ℝ,ℝ) satisfies f(0)=0 and the limits f∞=lim|s|→∞(f(s)/s), f0=lim|s|→0(f(s)/s)∈{0,∞}. Weight function a(t)∈C1[0,1] satisfies a(t)>0 on [0,1]

Persistence and Extinction Dynamics in Reaction-Diffusion-Advection Stream Population Model with Allee Effect Growth

The question how aquatic populations persist in rivers when individuals are constantly lost due to downstream drift has been termed the ``drift paradox. Reaction-diffusion-advection models have been used to describe the spatial-temporal dynamics of stream population and they provide some qualitative explanations to the paradox. Here random undirected movement of individuals in the environment is described by passive diffusion, and an advective term is used to describe the directed movement in a river caused by the flow. In this work, the effect of spatially varying Allee effect growth rate on the dynamics of reaction-diffusion-advection models for the stream population is studied. In the first part, a reaction-diffusion-advection equation with strong Allee effect growth rate is proposed to model a single species stream population in a unidirectional flow. Under biologically reasonable boundary conditions, the existence of multiple positive steady states is shown when both the diffusion coefficient and the advection rate are small, which lead to different asymptotic behavior for different initial conditions. On the other hand, when the advection rate is large, the population becomes extinct regardless of initial condition under most boundary conditions. It is shown that the population persistence or extinction depends on Allee threshold, advection rate, diffusion coefficient and initial conditions, and there is also rich transient dynamical behavior before the eventual population persistence or extinction. The dynamical behavior of a reaction-diffusion-advection model of a stream population with weak Allee effect type growth is studied in the second part. Under the open environment, it is shown that the persistence or extinction of population depends on the diffusion coefficient, advection rate and type of boundary condition, and the existence of multiple positive steady states is proved for intermediate advection rate using bifurcation theory. On the other hand, for closed environment, the stream population always persists for all diffusion coefficients and advection rates. In the last part, the dynamics of a reaction-diffusion-advection benthic-drift population model that links changes in the flow regime and habitat availability with population dynamics is studied. In the model, the stream is divided into drift zone and benthic zone, and the population is divided into two interacting compartments, individuals residing in the benthic zone and individuals dispersing in the drift zone. The benthic population growth is assumed to be of strong Allee effect type. The influence of flow speed and individual transfer rates between zones on the population persistence and extinction is considered, and the criteria of population persistence or extinction are formulated and proved. All results are proved rigorously using the theory of partial differential equation, dynamical systems. Various mathematical tools such as bifurcation methods, variational methods, and monotone methods are applied to show the existence of multiple steady state solutions of models