Role of Intra-specific Competition and Additional Food on Prey-Predator Systems exhibiting Holling Type-IV Functional Response

TL;DR

This work develops and analyzes a deterministic prey-predator model with a $Holling\,type-\!IV$ functional response, incorporating intra-specific predator competition and additional food. By nondimensionalizing the model, the authors examine positivity, boundedness, equilibria, and global dynamics, revealing rich bifurcation structures including transcritical, saddle-node, Hopf, and cusp bifurcations. They also formulate and solve time-optimal control problems for both the quality and quantity of additional food using the Pontryagin maximum principle, deriving switching conditions and illustrating results with CasADi-based simulations. The study provides insights into how provisioning of additional resources can drive pest eradication, coexistence, or pest dominance, offering practical guidance for bio-control and pest-management strategies.

Abstract

In recent years, the study on the impact of competition on additional food provided prey-predator systems have gained significant attention from researchers in the field of mathematical biology. In this study, we consider an additional food provided prey-predator model exhibiting Holling type-IV functional response and the intra-specific competition among 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 explore the codimension-$1$ and $2$ 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 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 (14)

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

Theorems & Definitions (13)

• Theorem 3.1
• proof
• Lemma 4.1
• Lemma 5.1
• Lemma 5.2
• Lemma 5.3
• Theorem 5.4
• Theorem 6.1
• proof
• Theorem 6.2