## Mathematical Analysis of Positone/Semipositone Models

### Overview:

In collaboration with various researchers, my work in the study of steady state solutions for reaction diffusion equations has involved use of/adaptation of various methods from nonlinear analysis such as the method of sub-super solutions, principle of linearized stability, time map analysis (quadrature methods), and degree theory to accommodate nonlinear boundary conditions present in the equations. In particular, I have studied the structure and/or stability of positive steady state solutions of reaction diffusion models arising from both combustion theory and population dynamics.

With regard to population dynamics, I have explored models with various growth terms including logistic and Allee effect, density dependent predation, and constant yield predation, all in the presence of nonlinear boundary conditions modeling conditional dispersal. An Allee effect is exhibited at the patch level when dynamics of the model predict persistence for some initial conditions but extinction for others, even with the same parameter ranges.

When the reaction diffusion equations include predation that is constant yield (i.e., not density dependent), the problem has what is known in the literature as “semipositone structure.” Proving existence of positive solutions of reaction diffusion equations with semipositone structure poses challenging mathematical problems.

### Publications:

#### with R. Shivaji

##### 2014, vol. 2014, no. 88, 1-27

We examine the structure of positive steady state solutions for a diffusive population model with logistic growth and negative density dependent emigration on the boundary. In particular, this class of nonlinear boundary conditions depends on both the population density and the diffusion coefficient. Results in the one-dimensional case are established via quadrature methods. Additionally, we discuss the existence of a Halo-shaped bifurcation curve.

#### with R. Shivaji

##### 2014, vol. 414, no. 2, 561-573

The structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined. In particular, a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary. Our existence results are established via the well-known sub-super solution method.

#### with Eun Kyoung Lee, Lakshmi Sankar, & R. Shivaji

##### 2013, vol. 2013, no. 97, 1-9

We consider the problem \begin{equation*} \label{eqn} \left\{ \begin{split} -\Delta u &=\frac{au-bu^2-c}{u^\alpha} , \quad x \in \Omega \\u &= 0, \qquad \qquad \qquad \ x \in \partial\Omega \end{split} \right. \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$, $a>0, b>0, c\geq0$ and $\alpha \in$ $(0, 1)$. Given $a,b$ and $\alpha,$ we establish the existence of a positive solution for small values of $c$. We also extend our results to the $\Delta_p$ operator and to corresponding exterior domain problems.

#### with K. Ashley & V. Sincavage

##### 2013, vol. 6, no. 4, 399-430

We examine a one-dimensional reaction diffusion model with a weak Allee growth rate that appears in population dynamics. We combine grazing with a certain nonlinear boundary condition that models negative density dependent dispersal on the boundary and analyze the effects on the steady states. In particular, we study the bifurcation curve of positive steady states as the grazing parameter is varied. Our results are acquired through the adaptation of a quadrature method and Mathematica computations. Specifically, we computationally ascertain the existence of Σ-shaped bifurcation curves with several positive steady states for a certain range of the grazing parameter.

#### with R. Shivaji

##### 2012, vol. 6, 150-157

We analyze the solutions of a population model with diffusion and strong Allee effect. In particular, we focus our study on a population that satisfies a certain nonlinear boundary condition and on its survival when constant yield harvesting is introduced. We discuss, in detail, results for the one-dimensional case.

#### with E. K. Lee & R. Shivaji

##### 2011, vol. 74, no. 17, 6202-6208

We consider a population model with diffusion, a strong Allee effect per capita growth function, and constant yield harvesting. In particular, we focus our study on a population living in a patch, $\Omega \subseteq \mathbb{R}^n$ with $n \geq 1$, that satisfies a certain nonlinear boundary condition. We establish our existence results by the method of sub-super solutions.

#### with E. K. Lee & R. Shivaji

##### 2011, vol. 375, no. 1, 365-370

We analyze the solutions of a population model with diffusion and logistic growth. In particular, we focus our study on a population living in a patch, $\Omega \subset \mathbb{R}^n$ with $n \geq 1$, that satisfies a certain non-linear boundary condition and on its survival when constant yield harvesting is introduced. We establish our existence results by the method of sub-super solutions.

#### with E. K. Lee & R. Shivaji

##### 2010, vol. 2010, no. 1, 365-370

We study the positive solutions to boundary value problems of the form \begin{eqnarray*} -\Delta u & = & \lambda f(u); \quad \Omega\\ \alpha(x, u)\frac{\partial u}{\partial \eta} & + & \left[1 - \alpha(x, u) \right]u = 0; \quad \partial \Omega \end{eqnarray*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with $n \geq 1$, $\Delta$ is the Laplace operator, $\lambda$ is a positive parameter, $f:[0, \infty) \longrightarrow (0, \infty)$ is a continuous function which is sublinear at $\infty$, $\frac{\partial u}{\partial \eta}$ is the outward normal derivative, and $\alpha(x, u):\Omega \times \mathbb{R} \longrightarrow [0, 1]$ is a smooth function nondecreasing in $u$. In particular, we discuss the existence of at least two positive radial solutions for $\lambda \gg 1$ when $\Omega$ is an annulus in $\mathbb{R}^n.$ Further, we discuss the existence of a double S-shaped bifurcation curve when $n = 1$, $\Omega = (0, 1)$, and $f(s) = e^{\frac{\beta s}{\beta + s}}$ with $\beta \gg 1.$

#### with E. K. Lee & R. Shivaji

##### 2010, Conf. 19, 135-149

We study a two point boundary-value problem describing the steady states of a Logistic growth population model with diffusion and constant yield harvesting. In particular, we focus on a model when a certain nonlinear boundary condition is satisfied.