Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

Abstract

This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.

A mathematical model of population dynamics revisited with fear factor, maturation delay, and spatial coefficients

“…Then, Equation () becomes $$$$ {\mathcal{A}}_t=\frac{1}{2}\left({q}_1y+\overline{q_1}\overline{y}\right){f}_n+W\left(y,\overline{y}\right), $$$$ with $$$ W\left(y,\overline{y}\right)=\frac{h}{2}\left(y+\overline{y}, iy-i\overline{y},0\right) $$$. Using the ideas from Wu and Zhao et al, 33,34 $$$ y $$$ satisfies $$$$ \dot{y}=i{\xi}_n\tilde{\eta}y+g\left(y,\overline{y}\right), $$$$ where $$$$ g\left(y,\overline{y}\right)=\left({\Omega}_1(0)-i{\Omega}_2(0)\right)\left\langle H\left({\mathcal{A}}_t,0\right),{f}_n\right\rangle . $$$$ …”