Dynamics and Control of Additional Food Provided Prey-Predator Systems exhibiting Holling Type-III Functional Response and Intra-specific Competition among Predators

TL;DR

The paper analyzes a two-species predator–prey model with Holling type‑III functional response, intra‑specific predator competition, and additional food. It combines positivity, boundedness, and equilibria analysis with detailed bifurcation study (transcritical, saddle‑node, Hopf) and reveals hysteresis in interior equilibria, as well as global dynamics across the added‑food parameter space. A time‑optimal control framework is developed for both the quality and quantity of added food, yielding bang–bang (and potential singular) strategies; CasADi-based numerical experiments illustrate pest‑management implications. Collectively, the results show how added food and predator competition influence pest suppression, bistability, and trajectories to desired states, informing ecologically safe and effective biocontrol strategies.

Abstract

The dynamics of predator-prey systems influenced by intra-specific competition and additional food resources have increasingly become a subject of rigorous study in the realm of mathematical biology. In this study, we consider an additional food provided prey-predator model exhibiting Holling type-III 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 possible bifurcations. We numerically depict the presence of Hysteresis loop in the system. 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. We show that the findings of these dynamics and control studies emphasises the role of additional food and intra-specific competition in bio-control of pests.

Figures (12)

• Figure 1: The possible configurations for the prey and predator nullclines of the system (\ref{['3iscd']}).
• Figure 2: Transcritical bifurcation diagram around axial equilibrium $E_1 = (\gamma,0)$ with respect to the quantity of additional food $\xi$.
• Figure 3: 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 4: Supercritical Hopf bifurcation diagram with respect to the intra-specific competition $\epsilon$.
• Figure 5: Nature of interior equilibria with respect to the intra-specific competition ($\epsilon$)

Theorems & Definitions (11)

• 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