Deterministic and Stochastic Studies on Additional Food Provided Prey-Predator Systems with Group Defence among Prey and Mutual Interference among Predators

TL;DR

This work analyzes a prey–predator system with Holling type‑IV group defence, mutual interference among predators, and additional food provisioning, extending the framework to stochastic settings and time‑optimal control. The authors derive the Holling type‑IV responses, nondimensionalize the model, and establish positivity, boundedness, and the existence of multiple equilibria, including an interior equilibrium whose stability depends on bifurcation structure such as transcritical, saddle‑node, and Hopf events. They map global dynamics in the α–ξ plane, identify regions of bistability, and formulate two time‑optimal control problems (varying food quality α or quantity ξ) solved via Pontryagin theory and numerical methods, showing reduced hitting times under control. The stochastic extension includes Brownian and Lévy noises with corresponding stochastic time‑optimal control, demonstrated through Monte Carlo simulations and statistical tests, underscoring practical implications for pest management and biocontrol strategies under uncertainty.

Abstract

The influence of competition and additional food on prey-predator dynamics has attracted considerable interest from mathematical biology researchers in recent times. In this study, we consider an additional food provided prey-predator model exhibiting Holling type-IV functional response among mutually interferring predators. We prove the existence and uniqueness of global positive solutions for the proposed model. We study the existence and stability of equilibrium points and further explored bifurcations with respect to the additional food and competition. We further study the global dynamics of the system and discuss the consequences of providing additional food. Later, we do the time-optimal control studies with respect to the quality and quantity of additional food as control variables by transforming the independent variable in the control system. Making use of the Pontraygin maximum principle, we characterize the optimal quality of additional food and optimal quantity of additional food. We further enhanced the model by incorpoating both continuous and discrete noise. We further characterized and numerically simulated the stochastic optimal controls through Sufficient Stochastic Maximum Principle. We show that the findings of these dynamics and control studies have the potential to be applied to a variety of problems in pest management.

Figures (16)

• Figure 2: The possible configurations for the prey and predator nullclines of the system (\ref{['4mid']}) when $\delta \xi - m (1+\alpha \xi) > 0$.
• Figure 3: The possible configurations for the prey and predator nullclines of the system (\ref{['4mid']}) when $0 > \delta \xi - m (1+\alpha \xi) > \frac{-(\delta - m)}{2 \sqrt{\omega}}$.
• Figure 4: Transcritical bifurcation diagram around trivial equilibrium $E_0 = (0,0)$ with respect to the quantity of additional food $\xi$.
• Figure 5: Transcritical bifurcation diagram around axial equilibrium $E_1 = (\gamma,0)$ with respect to the quantity of additional food $\xi$.
• Figure 6: Saddle-node bifurcation diagram around axial equilibrium $E_2 = \left(0,\frac{\delta \xi - m (1 + \alpha \xi)}{m \epsilon}\right)$ with respect to the quantity of additional food $\xi$.

Theorems & Definitions (15)

• Theorem 3.1
• proof
• Lemma 4.1
• Lemma 5.1
• Lemma 5.2
• Lemma 5.3
• Lemma 5.4
• Theorem 5.5
• Theorem 6.1
• proof