Existence of periodic solution of a non-autonomous allelopathic phytoplankton model with fear effect

TL;DR

The paper studies the existence of a positive $T$-periodic solution for a non-autonomous allelopathic phytoplankton model with fear effect. It reformulates the system via a log-transform $z_i= obreak \ln x_i$ to obtain a first-order operator equation and proves, using Mawhin's coincidence degree continuation theorem, that under conditions $(A1)$–$(A3)$ a $T$-periodic solution exists; $L$ is a Fredholm operator of index zero and $N$ is $L$-compact. An illustrative example with time-periodic parameters demonstrates the emergence of periodic dynamics, while the constant-coefficient case yields steady states, highlighting the role of temporal variability. The results provide a rigorous framework for seasonal dynamics in allelopathic phytoplankton and suggest avenues for extensions to time scales and fractional models, with potential applications to management of harmful algal blooms.

Abstract

In this paper, we consider a non-autonomous allelopathic phytoplankton competition ODE model, incorporating the influence of fear effects observed in natural biological phenomena. Based on Mawhin's coincidence degree theory some sufficient conditions for existence of periodic solutions are obtained. We validate our findings through an illustrative example and numerical simulations, showing that constant coefficients lead to steady-state dynamics, while periodic variations induce oscillatory behavior.

Figures (2)

• Figure 1: The periodic solution for system \ref{['e3.14']} with initial condition $x_1(0) = 0.002$ and $x_2(0) = 0.002$.
• Figure 2: Steady dynamics for system (3.14) for constant coefficient with initial condition $x_1(0) = 0.002$ and $x_2(0) = 0.002$.

Theorems & Definitions (9)

• Theorem 2.1: mawhin
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Example 3.1
• Remark 3.1