Maximising the total population in a diffusive predator-prey system using optimal control theory

Abstract

Studying population dynamics provides insights into the behavioural patterns of species. Several scholars have introduced different physical systems to understand behavioural patterns of real-world phenomena. Predator-prey interaction is one of the most common interrelationships in nature. Studying a diffusive predator-prey model is significant for several reasons which represents a more realistic representation of population dispersal in space, where a heterogeneous population exists. In this study, optimal control theory has been applied to a diffusive predator-prey system in analysing the population model to achieve the maximum total population. By implementing a time-dependent control variable 𝛼(𝑡) which is the mixing rate, to the diffusive predator-prey system, our concern is to maximise both predator and prey populations in a final time 𝑇. According to optimal control theory, under mathematical formulation, the Pay-off functional and the Hamiltonian are introduced along with the adjoint dynamics and the terminal conditions where the optimality criterion is satisfied. The Pontryagin Maximum Principle has been introduced such that the Hamiltonian is maximised where 0 ≤ 𝛼(𝑡) ≤ 1. More interestingly, we determined that the Hamiltonian would get its maximum value when 𝑝2𝜇2 - 𝑝1𝜇1 ≤ 0, where the optimal value of the control variable is 𝛼∗(𝑡) = 0 and if 𝑝2𝜇2 - 𝑝1𝜇1 > 0 then 𝛼∗(𝑡) = 1, where 𝑝1, 𝑝2 are costate functional and 𝜇1, 𝜇2 are the rate of prey loss and the growth rate of predators respectively. Furthermore, the term 𝑝2𝜇2 - 𝑝1𝜇1 is analysed under three cases which are 𝜇2 < 𝜇1, 𝜇2 = 𝜇1 and 𝜇2 > 𝜇1. From the analytical approach, we obtained that for the cases 𝜇2 < 𝜇1 and 𝜇2 = 𝜇1 the system is uncontrollable, and a switching time does not exist. The study reveals that the system is considered controllable only when 𝜇2 > 𝜇1 where a switching time exists such that the total population is maximised. A simulation is conducted under the case 𝜇2 > 𝜇1 to depict the results, achieving the maximum population at 𝑡 = 16.5 while setting the switching time as 𝑡 = 11. Finally, the obtained results were validated. Therefore, using both numerical simulation and estimates we concluded that 𝛼∗(𝑡) = 0 when [0, 𝑡2] and 𝛼∗(𝑡) = 1 when (𝑡2, 𝑇], where 𝑡2 is the switching time. In future work, it is expected to improve the analysis by implementing both time and space dependent control variable 𝛼(𝑥, 𝑡).