Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients

Abstract

Discarded plastic is subjected to weather effects from different ecosystems and becomes microplastic particles. Due to their small size, they have spread across the planet. Their presence in living organisms can have several harmful consequences, such as altering the interaction between prey and predator. Huang et al. successfully modeled this system presenting numerical results of ecological relevance. Here, we have rewritten their equations and solved a set of them analytically, confirming that microplastic particles accumulate faster in predators than in prey and calculating the time values from which it happens. Using these analytical solutions, we have retrieved the Lotka–Volterra predator–prey model with time-varying intraspecific coefficients, allowing us to interpret ecological quantities referring to microplastics dispersion. After validating our equations, we solved analytically particular situations of ecological interest, characterized by extreme effects on predatory performance, and proposed a second-order differential equation as a possible next step to address this model. Our results open space for further refinement in the study of predator–prey models under the effects of microplastic particles, either exploring the second-order equation that we propose or modify the Huang et al. model to reduce the number of parameters, embedding in the time-varying intraspecies coefficients all the adverse effects caused by microplastic particles.

Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis

“…In this paper, we introduce an impulsive Lotka-Volterra-type models using the conformable calculus approach. The introduced model extends and complements numerous existing integer-order Lotka-Volterra models [1][2][3][4][5][6], impulsive Lotka-Volterra models [27,29,30,32], as well as, Lotka-Volterra models with classical fractional-order derivatives [16][17][18][19][20]34] to the impulsive conformable case. The benefits of the conformable derivatives make the introduced model more relevant to the real-world applications.…”

“…Remark 3. The above model generalizes numerous existing integer-order Lotka-Volterra models [1][2][3][4][5][6], impulsive Lotka-Volterra models [27,29,30,32], as well as, Lotka-Volterra models with classical fractional-order derivatives [16][17][18][19][20]34] to the impulsive conformable case. In fact, the use of generalized conformable derivatives is motivated by their advantages in applications related to the simplifications in the use of the chain rule.…”

“…Among them, the Lotka-Volterra competitive systems are one of the most studied phenomena since they generalize several predator-prey models [1][2][3]. Different classes of Lotka-Volterra models has been proposed and investigated in the existing literature, including systems with dispersion [4][5][6][7].…”