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Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions

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The third author was supported by the NSF award DMS- 1853352 to perform this research work

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  • We study the structure of positive solutions to steady state ecological models of the form:

    $ \begin{array}{l} \left\{ \begin{split} -\Delta u& = \lambda uf(u)\; \; && {\rm{in}}\; \; \Omega,\\ \alpha(u)&\frac{\partial u}{\partial \eta}+[1-\alpha(u)]u = 0 &&\;\;\;{\rm{on}}\; \; \partial\Omega, \end{split} \right. \end{array} $

    where $ \Omega $ is a bounded domain in $ \mathbb{R}^n; $ $ n>1 $ with smooth boundary $ \partial\Omega $ or $ \Omega = (0,1) $, $ \frac{\partial}{\partial\eta} $ represents the outward normal derivative on the boundary, $ \lambda $ is a positive parameter, $ f:[0,\infty)\to \mathbb{R} $ is a $ C^2 $ function such that $ \tfrac{f(s)}{k-s}>0 $ for some $ k>0 $, and $ \alpha:[0,k]\to[0,1] $ is also a $ C^2 $ function. Here $ f(u) $ represents the per capita growth rate, $ \alpha(u) $ represents the fraction of the population that stays on the patch upon reaching the boundary, and $ \lambda $ relates to the patch size and the diffusion rate. In particular, we will discuss models in which the per capita growth rate is increasing for small $ u $, and models where grazing is involved. We will focus on the cases when $ \alpha'(s)\geq 0 $; $ [0,k] $, which represents negative density dependent dispersal on the boundary. We employ the method of sub-super solutions, bifurcation theory, and stability analysis to obtain our results. We provide detailed bifurcation diagrams via a quadrature method for the case $ \Omega = (0,1) $.

    Mathematics Subject Classification: Primary: 35K57; Secondary: 35B32, 92D40.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 1-\frac{s}{10}-1.5\frac{s}{1+s^2} $ with $ \alpha(1) = 1 $.

    Figure 2.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 1-\frac{s}{10}-1.5\frac{s}{1+s^2} $ with $ \alpha(1)<1 $.

    Figure 3.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 5-s-\frac{1}{1+5s} $ with $ \alpha(1) = 1 $.

    Figure 4.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 5-s-\frac{1}{1+5s} $ with $ \alpha(1)<1 $.

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