A Nonlinear Diffusion Equation in Phytoplankton Dynamics with Self-Shading Effect

Abstract

We consider the nonlinear diffusion equation

$${p_{t}}={p_{{xx}}}-\omega {p_{x}}-\lambda p+f\left({I\left(p\right)} \right)p in\left({0,\ell}\right)\times\left({0,\infty}\right),$$

((1.1))

where

$$I\left( p\right)\equiv I\left(p\right)\left({x,t}\right)=\int\limits_{0}^{x}{\left({k+p\left({\xi ,t}\right)}\right)}d\xi.$$

Here κ ≥ 0, λ > 0, ω ≥ 0, and 0 < ℓ ≤ ∞ are given constants, f: [0, ∞) → (0, ∞) is a decreasing Lipschitz function such that lim_r→∞ f(r) = 0, and p: [0, ℓ] × [0, ∞) → [0, ∞) is the unknown. Imposed are the boundary condition

$${p_{x}}\left({0,t}\right)=\omega p(0,t)$$

((1.2))

,

$$\left\{{_{{\mathop{{\lim }}\limits_{{x \to\infty}}p\left({x,t}\right)= 0if\ell=\infty }}^{{{p_{x}}\left({\ell,t}\right)=\omega p\left({\ell,t}\right)ifell< \infty,}}}\right.$$

((1.3))

for all t > 0 and the initial condition

$$p=\left({x,0}\right)={p_{0}}\left(x\right)forx \in\left({0,\ell}\right).$$

((1.4))