Modeling the effects of trait-mediated dispersal on coexistence of mutualists

© 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) Even though mutualistic interactions are ubiquitous in nature, we are still far from making good predictions about the fate of mutualistic communities under threats such as habitat fragmentation and climate change. Fragmentation often causes declines in abundance of a species due to increased susceptibility to edge effects between remnant habitat patches and lower quality “matrix” surrounding these focal patches. It has been argued that ecological communities are replete with trait-mediated indirect effects, and that these effects may sometimes contribute more to the dynamics of a population than direct density-mediated effects, e.g., lowering an organism\u27s fitness through competitive interactions. Although some studies have focused on trait-mediated behavior such as trait-mediated dispersal, in which an organism changes its dispersal patterns due to the presence of another species, they have been mostly limited to predator-prey systems-little is known regarding their effect on other interaction systems such as mutualism. Here, we explore consequences of fragmentation and trait-mediated dispersal on coexistence of a system of two mutualists by employing a model built upon the reaction diffusion framework. To distinguish between trait-mediated dispersal and density-mediated effects, we isolate effects of trait-mediated dispersal on the mutualistic system by excluding any direct density-mediated effects in the model. Our results demonstrate that fragmentation and trait-mediated dispersal can have important impacts on coexistence of mutualists. Specifically, one species can be better able to invade and persist than the other and be crucial to the success of the other species in the patch. Matrix quality degradation can also bring about a complete reversal of the role of which species is supporting the other\u27s persistence in the patch, even as the patch size remains constant. As most mutualistic relationships are identified based on density-mediated effects, such an effect may be easily overlooked

An infinite semipositone problem with a reversed S-shaped bifurcation curve

We study positive solutions to the two point boundary value problem: \begin{equation*} \begin{matrix}Lu = -u'' = \lambda \bigg\{\dfrac{A}{u^\gamma}+M\big[u^\alpha+u^\delta\big]\bigg\} \; ;\; (0, 1) \\ u(0) = 0 = u(1)\; \; \; \; \; \; \; \; \; \; \end{matrix} \end{equation*} where A < 0 , \alpha \in (0, 1), \delta > 1, \gamma \in (0, 1) are constants and \lambda > 0, M > 0 are parameters. We prove that the bifurcation diagram $(\lambda \text{ vs } \|u\|_\infty)$ for positive solutions is at least a reversed S-shaped curve when $M\gg1$. Recent results in the literature imply that for $M\gg1$ there exists a range of $\lambda$ where there exist at least two positive solutions. Here, when $M\gg1$, we prove the existence of a range of $\lambda$ for which there exist at least three positive solutions and that the bifurcation diagram is at least a reversed S-shaped curve. Further, via a quadrature method and Python computations, for $M\gg1$, we show that the bifurcation diagram is exactly a reversed S-shaped curve. Also, when the operator $L$ is replaced by a $p$-Laplacian operator with p > 1 , as well as $p$-$q$ Laplacian operator with $p = 4$ and $q = 2$, we show that the bifurcation diagram is again an exactly reversed S-shaped curve when $M\gg1$

Existence results for a class of pp–qq Laplacian semipositone boundary value problems

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$; $N>1$ with a smooth boundary or $\Omega=(0,1)$. We study positive solutions to the boundary value problem of the form: \begin{equation*} \begin{aligned} -\Delta_p u - \Delta_q u&=\lambda f(u) &&\mbox{in}~\Omega,\\ u &= 0 &&\mbox{on}~\partial\Omega, \end{aligned} \end{equation*} where $q \in [2,p)$, $\lambda$ is a positive parameter, and $f:[0,\infty) \mapsto \mathbb{R}$ is a class of $C^1$, non-decreasing and $p$-sublinear functions at infinity (i.e. $\lim_{t \rightarrow \infty} \frac{f(t)}{t^{p-1}}=0$) that are negative at the origin (semipositone). We discuss the existence of positive solutions for $\lambda\gg1$. Further, when $p=4,q=2$, $\Omega=(0,1)$ and $f(s)=(s+1)^\gamma-2$; $\gamma \in (0,3)$, we provide the exact bifurcation diagram for positive solutions. In particular, we observe two positive solutions for a finite range of $\lambda$ and a unique positive solution for $\lambda\gg1.

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

Analysis of positive solutions for classes of nonlinear reaction diffusion equations and systems

The focus of this thesis is to study positive solutions for classes of nonlinear reaction diffusion equations and systems. In particular, we consider three focuses. In Focus 1, we establish existence and uniqueness of positive solutions for a class of infinite semipositone problems with nonlinear boundary conditions. In Focus 2, we explore the consequences of fragmentation and trait-mediated dispersal on the coexistence of a system of two mutualists by employing a model built upon the reaction diffusion framework. We establish several coexistence and nonexistence results. Finally, in Focus 3, we develop and analyze a radial finite difference method that directly approximate solutions for classes of semipositone problems with Dirichlet boundary conditions. Our existence results in Focus 1 and Focus 2 are achieved by methods of sub and supersolutions. In Focus 3, via computational methods, we obtain bifurcation diagrams describing the structure of positive solutions. Namely, we obtain these bifurcation diagrams via a modified finite difference method and MATLAB computations in the case when the domain is the unit disc in R2. This dissertation aims to significantly enrich the mathematical and computational analysis literature on reaction diffusion equations and systems. [This abstract may have been edited to remove characters that will not display in this system. Please see the PDF for the full abstract.]]]> 2021 Differential equations, Nonlinear $x Numerical solutions Reaction-diffusion equations$x Numerical solutions Nonlinear systems $x Mathematical models Diffusion$x Mathematical models English http://libres.uncg.edu/ir/uncg/f/Muthunayake_uncg_0154D_13344.pdf oai:libres.uncg.edu/36133 2021-09-07T13:26:54Z UNCG Empathy after entropy: Crane, Toomer, Woolf, and O’Neill NC DOCKS at The University of North Carolina at Greensboro Phillips, Matthew Michael <![CDATA[This dissertation is about the effects of entropy within various literary texts written in English. My main argument is that an increase in the overall entropy within the closed system, so to speak, of a text directly affects the empathic capacity of the characters within it. In general, if the entropy within a text increases, the empathic capacity of one or more of its characters increases. Sometimes, the text’s setting has reached a point of near-stasis, a situation in which entropy is already nearing its maximum. In such cases, there is often a character who experiences the last of a dwindling entropic potential, and this encounter is crucial regarding the remaining corresponding empathic potential. I rely heavily on close reading, as well as on biographical and interdisciplinary research to show how increased entropy results in the erosion of barriers between characters which previously appeared unbreachable

Modeling the effects of trait-mediated dispersal on coexistence of mutualists

Even though mutualistic interactions are ubiquitous in nature, we are still far from making good predictions about the fate of mutualistic communities under threats such as habitat fragmentation and climate change. Fragmentation often causes declines in abundance of a species due to increased susceptibility to edge effects between remnant habitat patches and lower quality "matrix" surrounding these focal patches. It has been argued that ecological communities are replete with trait-mediated indirect effects, and that these effects may sometimes contribute more to the dynamics of a population than direct density-mediated effects, e.g., lowering an organism's fitness through competitive interactions. Although some studies have focused on trait-mediated behavior such as trait-mediated dispersal, in which an organism changes its dispersal patterns due to the presence of another species, they have been mostly limited to predator-prey systems-little is known regarding their effect on other interaction systems such as mutualism. Here, we explore consequences of fragmentation and trait-mediated dispersal on coexistence of a system of two mutualists by employing a model built upon the reaction diffusion framework. To distinguish between trait-mediated dispersal and density-mediated effects, we isolate effects of trait-mediated dispersal on the mutualistic system by excluding any direct density mediated effects in the model. Our results demonstrate that fragmentation and trait-mediated dispersal can have important impacts on coexistence of mutualists. Specifically, one species can be better able to invade and persist than the other and be crucial to the success of the other species in the patch. Matrix quality degradation can also bring about a complete reversal of the role of which species is supporting the other's persistence in the patch, even as the patch size remains constant. As most mutualistic relationships are identified based on density-mediated effects, such an effect may be easily overlooked