On the solutions of the two preys and one predator type model approached by the fixed point theory
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Sådhanå (2020)45:211 https://doi.org/10.1007/s12046-020-01468-1
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On the solutions of the two preys and one predator type model approached by the fixed point theory ALI TURAB and WUTIPHOL SINTUNAVARAT* Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12120, Thailand e-mail: [email protected] MS received 14 December 2019; revised 30 March 2020; accepted 7 May 2020 Abstract. The purpose of this paper is to discuss a special type of functional equation that describes the relationship between the predator animals and their two choices of prey with their corresponding probabilities. Our aim is to find the existence and uniqueness results of the proposed functional equation using the Banach fixed point theorem. Finally, we give an illustrative example that supports our main results. Keywords. Functional equations; mathematical biology; predator–prey model; fixed points; Banach fixed point theorem.
1. Introduction The vigorous link of the predator and their prey is one of the major concerns in the ecosystem. Recently, researchers found that predation can impact the size of the prey population by going about as top-down control. Indeed, the interaction between these two types of population control provides opportunities to observe the changes in population over time [1, 2]. In 1973, Lyubich and Shapiro [3] studied the existence and uniqueness of a continuous solution u : ½0; 1 ! ½0; 1 of the following functional equation: uðxÞ ¼ xuðð1 lÞx þ lÞ þ ð1 xÞuðð1 mÞxÞ; x 2 ½0; 1; ð1:1Þ where 0\l m\1: The functional equation (1.1) appears in mathematical biology and the theory of learning to observe the nature of predator animals that hunt two kinds of prey. Such a conduct is defined by the Markov process in the state space [0, 1] with the probabilities of transition given by Pðx ! ð1 lÞx þ lÞ ¼ x; Pðx ! ð1 mÞxÞ ¼ 1 x: In the mathematical model (1.1) the solution u is the final probability of the event when the predator is fixed on one category of prey, knowing that the initial probability for this category to be chosen is equal to x. Also, Turab and Sintunavarat [4] used such a type of functional equation to
observe the behaviour of the paradise fish in a two-choice situation. In [3], Lyubich and Shapiro used Schauder’s fixed point theorem to prove the existence of a solution of the functional equation (1.1) of the following form: uðxÞ ¼
1 P
ji xi ; ji 0;
i¼1
satisfying the conditions uð0Þ ¼ 0; uð1Þ ¼ 1:
After this, Istra˘¸t escu [5] proposed the existence and uniqueness result for the solution of the functional equation (1.1) with condition (1.2) using the Banach contraction mapping principle. In this context, Dmitriev and Shapiro [6] found a solution of (1.1) by a direct method. They denoted k1 ¼ 1 l and k2 ¼ 1 m and used the substitution uðxÞ :¼ x þ ðk2 k1 Þxð1 xÞfðxÞ;
ð1:3Þ
to reduce the functional equation (1.1) with the unknown function u to the f
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