Abstract
The aim of this paper is to investigate the dynamical behavior of the following model which describes the logistic difference equation taking into account the subjectivity in the state variables and in the parameters. where is a sequence of positive fuzzy numbers. and the initial value are positive fuzzy numbers. The existence and uniqueness of the positive solution and global asymptotic behavior of all positive solution of the fuzzy logistic difference equation are obtained. Moreover, some numerical examples are presented to show the effectiveness of results obtained.
1. Introduction
The first models for growing population were the classical Malthus and Verhulst (or logistic) models which deal with populations with one species [1]. In these models, the identification of the parameters is usually based on statistical methods, starting from data experimentally obtained and on the choice of some method adapted to the identification. These models are often subjected to inaccuracies (fuzzy uncertainty) that can be caused by the nature of the state variables, by parameters as coefficients of the model and by initial conditions.
In our real life, scientists have accepted the fact that uncertainty is very important study in most applications and they also have learned how to deal with uncertainty. Modeling the real life problems in such cases usually involves vagueness or uncertainty in some of the parameters or initial conditions. It is well known that fuzzy set introduced by Zadeh [2] is one of suitable tools and its development has been growing rapidly to various situations of theory and application including the theory of differential equations and difference equations with uncertainty. The latter is known as fuzzy difference equation whose parameters or the initial values are fuzzy numbers, and its solution is a sequence of fuzzy numbers. It has been used to model a dynamical system under possibility uncertainty [3].
To the best of our knowledge, the behavior of the parametric fuzzy difference equation is not always the same as the behavior of corresponding parametric ordinary difference equation. In recent decades, there is an increasing interest in studying fuzzy difference equation by many scholars. Some results concerning the study of fuzzy difference equations are included in these papers (see, for example, [3–18]).
What we propose in this paper is, to some extent, a generalization of classical logistic discrete model, using the subjectivity which comes from “fuzziness” of the biological phenomenon. The main aim in this paper is to investigate the dynamical behavior of the following logistic discrete time system:where parameter and the initial condition are positive fuzzy numbers.
The rest of this paper is organized as follows. In Section 2, we introduce some definitions and preliminaries. In Section 3, we study the existence, uniqueness, and global asymptotic behavior of the positive fuzzy solutions to system (1). Some numerical examples are given to show effectiveness of results obtained in Section 4. A general conclusion is drawn in Section 5.
2. Mathematical Preliminaries
To be convenience, we give some definitions used in the sequel.
Definition 1 (see [3]). is said to be a fuzzy number if satisfies (i)-(iv).
(i) is normal; i.e., there exists an such that .
(ii) is fuzzy convex; i.e., for all and such that (iii) is upper semicontinuous.
(iv) The support of , is compact, where denotes the closure of .
Let be the set of all real fuzzy numbers which are normal, upper semicontinuous, convex, and compactly supported fuzzy sets.
Definition 2 (fuzzy number (parametric form) [19]). A fuzzy number in a parametric form is a pair of functions , which satisfies the following requirements:
(1) is a bounded monotonic increasing left continuous function.
(2) is a bounded monotonic decreasing left continuous function.
(3)
A crisp (real) number is simply represented by The fuzzy number space becomes a convex cone which could be embedded isomorphically and isometrically into a Banach space [17].
Definition 3 (see [19]). Let , and arbitrary Then
(i) iff
(ii)
(iii)
(iv) (v) ,
Definition 4 (triangular fuzzy number [19]). A triangular fuzzy number (TFN) denoted by is defined as where the membership function
The cuts of are denoted by , ; it is clear that is a closed interval. A fuzzy number is positive if
The following proposition is fundamental since it characterizes a fuzzy set through the -levels.
Proposition 5 (see [19]). If is a compact, convex, and not empty subset family of such that
(i)
(ii) if
(iii) if
then there is such that for all and
Definition 6 (see [4]). A sequence of positive fuzzy numbers persists (resp., is bounded) if there exists a positive real number (resp., ) such thatA sequence of positive fuzzy numbers is bounded and persists if there exist positive real numbers such that
Definition 7. is called a positive solution of (1), if is a sequence of positive fuzzy numbers which satisfies (1). The equilibrium of (1) is the solution of the following equation:
Definition 8 (see [3]). Let be fuzzy numbers with ; then the metric of and is defined as
Definition 9. Let be a sequence of positive fuzzy numbers and is a positive fuzzy number. Suppose thatandThe sequence converges to with respect to as if .
Definition 10. Let be a positive equilibrium of (1). The positive equilibrium is stable, if, for every , there exists a such that for every positive solution of (1), which satisfies , we have for all The positive equilibrium is asymptotically stable, if it is stable and every positive solution of (1) converges to the positive equilibrium of (1) with respect to as
3. Main Results
First we study the existence of the positive solutions of (1). We need the following lemma.
Lemma 11 (see [20]). Let be continuous; and are fuzzy numbers. Then
Theorem 12. Consider (1) where are positive fuzzy numbers. Then, for any positive fuzzy numbers , there exists a unique positive solution of (1).
Proof. The proof is similar to Proposition 2.1 [5]. Suppose that there exists a sequence of fuzzy numbers satisfying (1) with initial condition . Consider -cuts, , applying Lemma 11; we haveFrom (11), we can get the following system of ordinary difference equation with parameter , for ,Then, for any initial condition , there exists a unique solution
Now we show that , determines the solution of (1) with initial value , where is the solution of system (12) with initial conditions , satisfyingSince are positive fuzzy numbers, for , and, from Definition 2, then we haveWe claim that, for ,Inductively, it is clear that (15) is true for . Suppose that (15) holds true for . Then, from (12), (14), and (15) for , it follows that Therefore, (15) is satisfied.
Moreover, for , it follows from (12) that Since are positive fuzzy numbers, by Definition 2, then we have that are left continuous. From (17) we have that are left continuous. By induction we can get that are left continuous.
Next we prove that the support of , is compact. It is sufficient to prove that is bounded.
Let ; since and are positive fuzzy numbers, there exist constants such that, for , Hence, from (17) and (18), for , we get It is clear thatTherefore, (20) implies is compact, and Deducing inductively it can follow easily thatTherefore, from (15), (21), and being left continuous, it can be concluded that determines a sequence of positive fuzzy numbers satisfied with (13).
We prove that is a solution of (11) with initial value , since, for , Namely, is a solution of (11) with initial value .
Suppose that there exists another solution of (11) with initial value . Then from arguing as above we can easily get that, for ,Then from (13) and (23), we have , and hence This completes the proof of Theorem 12.
In order to study the dynamical behavior of the solution to (1), we first consider the following system of difference equations It is clear that the equilibrium points of (24) include the following four cases:
Lemma 13. Consider the system of difference equations (24), where , and are positive real constants, and the initial values are positive real numbers; then the following statements are true.
(i) The equilibrium is local asymptotically stable if .
(ii) System (24) has infinite numbers equilibrium which is unstable if .
(iii) System (24) has infinite numbers equilibrium which is unstable if .
(iv) Suppose that ; then system (24) has a unique positive equilibrium which is unstable.
Proof. (i) It is clear that is always equilibrium. We can easily obtain that the linearized system of (24) about the positive equilibrium isfrom which we can easily obtain that the eigenvalue . Since and , so all of eigenvalue lie inside the unit disk. This implies that the equilibrium is local asymptotically stable.
(ii) We can obtain that the linearized system of (24) about the positive equilibrium iswhereThe character equation of the linearized system of (28) about equilibrium is It is clear that the characteristic root of (30) has a root . Hence the equilibrium is unstable.
(iii) The proof of (iii) is similar to the proof of (ii). So it is omitted.
(iv) We can obtain that the characteristic equation of the linearized system of (28) about equilibrium is The character root of (31) is . It is clear that there exists a character root larger than 1. Hence the equilibrium is unstable.
Theorem 14. Consider the fuzzy difference equation (1). Suppose that there exists positive constants , for all such thatThen
Proof. Let be a positive solution of (1) with initial conditions ; applying (i) of Lemma 13, we get that , satisfies the following family of systems of parametric ordinary difference equationsFrom (34), we have that for all Since , it follows from (35) that for all Therefore,
Theorem 15. Suppose that and 1 are positive real numbers (trivial fuzzy numbers) and there exists a constant such that Then (1) has a unique positive equilibrium such that
Proof. Suppose that there exists a fuzzy number such that where Then from (39) we can obtain that Hence we can easily obtain that is a unique positive solution of (40).
Conversely, we prove that , where is the unique positive solution of (40) satisfying (38), determining a fuzzy number which satisfies (39). From (38) we get, for ,Then from (41) for all we haveMoreover, since is a fuzzy number, we have that are left continuous. Then from (37) and (38) we have that are left continuous.
Furthermore since (37) holds and is a positive fuzzy number, there exists a constant such that . Then from (37) and (38) we obtain that from which we have thatTherefore, from Theorem 2.1 of [13], (42), (44), and being left continuous we have that is a positive fuzzy number which satisfies (39). This completes the proof of Theorem 15.
Theorem 16. Consider fuzzy difference equation (1); if two positive fuzzy numbers are all real number 1 (trivial fuzzy numbers) and the initial value is a positive fuzzy number, suppose that there exists a positive constant , such that Then the solution of system (1) converges to .
Proof. By Theorem 12, we have that a sequence of a positive fuzzy number with is the solution of (1), satisfying From (45) and (46), it follows that We can prove inductively that From (46), it is easy to get Namely, .
Remark 17. From Theorem 16, we can conclude that the dynamical behavior of (1) is correlated with the initial value . When the initial value is a positive fuzzy number which is smaller than one, no matter how long the support of is, the solution of fuzzy difference equation (1) eventually converges to . From a biological point of view, if the population initial value is too small, even if the growth rate of the population is high (equal one), the population eventually becomes extinct.
4. Some Illustrative Examples
In order to illustrate our results obtained, we give some numerical examples to show effectiveness of results.
Example 18. Consider discrete fuzzy logistic system with initial value : where and the initial value are positive fuzzy numbers such that From that, we get, for , From (50) and (52), it results in a coupled system of difference equation with parameter , It is clear that (32) of Theorem 14 is satisfied. Therefore, the solution of (50) converges to 0. (see Figures 1–3)
Example 19. Consider discrete fuzzy logistic system (50) with initial value where and initial value .
It results in a coupled system of difference equations with parameter Clearly, all conditions of Theorem 16 are satisfied. Therefore, the solution of (50) converges to 0 (see Figures 4–6).
5. Conclusion
This paper deals with the dynamical behavior of single population logistic model under fuzzy environment. Firstly, the existence of positive fuzzy solution of this model is proved. Secondly, we obtained the following results
(i) If , then the positive solution of (1) eventually converges to 0 no matter how much the number of population initial values are.
(ii) If the number of population model is too small, even if the growth rate of population is 1, the population also converges to 0 (extinct).
(iii) When the parameter of model , the system has unique positive equilibrium .
Finally, some examples are presented to show effectiveness of results.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (Grant no. 11761018) and Key Research Project of Guizhou University of Finance and Economics (2018XZD02).