Population dynamics with symmetric and asymmetric harvesting

J. Ali, Florida Gulf Coast University, U.S.A.
D. Perry, New York University, U.S.A
S. Sasi, Mississippi State University, MS, U.S.A.
J. Schaefer, Northern Arizona University, U.S.A.
B. Schilling, Mississippi State University, MS, U.S.A.
R. Shivaji, Mississippi State University, MS, U.S.A.
M. Williams, Clarkson State University, U.S.A.

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 2., pp. 1-16.

Abstract: We study the positive solutions to steady state reaction diffusion
equations with Dirichlet boundary conditions of the forms:
\begin{align}
-u''&=\left\{\begin{array}{ll}
\lambda[au-bu^{2}-c], & x\in(L,1-L),\\
\lambda[au-bu^{2}], & x\in(0,L)\cup(1-L,1),
\end{array} \right.\tag{A}\\
u(0)&=0 =u(1),\nonumber
\end{align}
and
\begin{align}
-u''&=\left\{\begin{array}{ll}
\lambda[au-bu^{2}-c], & x\in(0,\frac{1}{2}),\\
\lambda[au-bu^{2}], & x\in(\frac{1}{2},1),
\end{array} \right.\tag{B}\\
u(0)&=0 =u(1).\nonumber
\end{align}
Here $\lambda, a, b, c$ and $L$ are positive constants with $0<L<\frac{1}{2}$. Such steady state equations arise in population dynamics with logistic type growth and constant yield harvesting. Here $u$ is the population density, $\frac{1}{\lambda}$ is the diffusion coefficient and $c$ is the harvesting effort. In particular, model A corresponds to a symmetric harvesting case and model B to an asymmetric harvesting case. Our objective is to study the existence of positive solutions and also discuss the effects of harvesting. We will develop appropriate quadrature methods via which we will establish our results.

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