Shadow system approach to a plankton model generating harmful algal bloom

Abstract

Spatially localized blooms of toxic plankton species have negative impacts on other organisms via the production of toxins, mechanical damage, or by other means. Such blooms are nowadays a worldwide spread environmental issue. To understand the mechanism behind this phenomenon, a two-prey (toxic and nontoxic phytoplankton)-one-predator (zooplankton) Lotka-Volterra system with diffusion has been considered in a previous paper. Numerical results suggest the occurrence of stable non-constant equilibrium solutions, that is, spatially localized blooms of the toxic prey. Such blooms appear for intermediate values of the rate of toxicity $μ$ when the ratio $D$ of the diffusion rates of the predator and the two prey is rather large. In this paper, we consider a one-dimensional limiting system (we call it a shadow system) in $(0,L)$ as $D \to \infty $ and discuss the existence and stability of non-constant equilibrium solutions with large amplitude when $μ$ is globally varied. We also show that the structure of non-constant equilibrium solutions sensitively depends on $L$ as well as $μ$.