Abstract
This work is concerned with the qualitative behavior of discrete time single species model with fuzzy environment where denotes the number of individuals of generation , is the intrinsic growth rate, and is interpreted as the carrying capacity of the surrounding environment. is a sequence of positive fuzzy number. and the initial value are positive fuzzy numbers. Applying difference of Hukuhara (H-difference), the existence, uniqueness of the positive solution, and global asymptotic behavior of all positive solution with the model are obtained. Moreover a numerical example is presented to show the effectiveness of theoretic results obtained.
1. Introduction
Difference equations or discrete time dynamical systems have many applications in economics, biology, computer science, control engineering, etc. (see, for example, [1–10] and the references therein). In the recent years, many researchers pay a close attention to the study on qualitative behavior of difference equation in mathematical biology and population dynamics [11–13]. In theoretical ecology, the difference equation models are often described the interactions of species with nonoverlapping generations. For example, in 1954, Ricker [14] forecasted fish stock recruitment using the following discrete time single species modelwhere denotes the number of individuals of generation , is the intrinsic growth rate, and is interpreted as the carrying capacity of the surround environment. In fact, from a biological point of view, researchers focus on whether or not all species in a multispecies community can be permanent or bounded.
As is all known, the parameters of the model are usually based on statistical method or on the choice of some method adapted to the identification. Therefore, these models are subjected to inaccuracies (fuzzy uncertainty) that can be caused by the nature of the state variables, by coefficients of the model and by initial conditions. In our real life, scientists are concerned with uncertainty and accept the fact that uncertainty is very important influencing factor of dynamical behavior of dynamical system. Fuzzy set introduced by Zadeh [15] and its development have been growing rapidly to various situation of theory and application including the theory of differential and difference equations with uncertainty. It is well known that a fuzzy difference equation is a difference equation where parameters or the initial values of systems are fuzzy numbers, and its solutions are sequences of fuzzy numbers.
In fact, the dynamical behavior of fuzzy difference equation is different from the behavior of corresponding parametric ordinary difference equation. In recent decades, researchers have an increasing interest in studying fuzzy difference equation. Some results on fuzzy difference equations have been reported (see, for example, [16–31]). Barros, Bassanezi, and Tonelli [32] have investigated the dynamical behavior of population model with fuzzy uncertainty. However, to the best of our knowledge, few authors study discrete time single species model under fuzzy environment. This paper is the first to study the dynamical behavior of discrete time single species model using fuzzy sets theory.
The main aim of this paper is to study the dynamical behaviors of the following discrete time single species modelwhere denotes the number of individuals of generation and is a sequence of positive fuzzy numbers. , are the intrinsic growth rate and the carrying capacity of the surround environment, respectively. Parameters and the initial condition are positive fuzzy numbers. This paper is, to some extent, a generalization of classic discrete time single species model, using the subjectivity which comes from “fuzziness” of the biological phenomenon.
The rest of this paper is organized as follows. In the next section, we introduce some definitions and preliminaries. In Section 3, the dynamical behaviors on the existence, uniqueness, and global asymptotic behaviors of the positive fuzzy solution to system (2) are studied. A numerical example is given to show effectiveness of results obtained in Section 4. Finally, a general conclusion is drawn in Section 5.
2. Mathematical Preliminaries
For convenience, we give some definitions used in the sequel.
Definition 1 (see [33]). is said to be a fuzzy number if satisfies the below (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) [34]). A fuzzy number in a parametric form is a pair of function , 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. [26]
Definition 3. Let , and arbitrary Then
(i) iff ,
(ii) ,
(iii)(iv) = , .
Definition 4. Let , and if there exists such that , then is called the H-difference of and and it is denoted by
In this paper the “−” sign stands always for H-difference and let us remark that .
Definition 5 (triangular fuzzy number [34]). A triangular fuzzy number (TFN) denoted by is defined as where the membership function
The cuts of are denoted by = = , , and it is clear that the are closed interval. A fuzzy number is positive if
The following proposition is fundamental since it characterizes a fuzzy set through the -levels.
Proposition 6 (see [33]). 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 7 (see [27]). 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 8. is a positive solution of (2), if is a sequence of positive fuzzy numbers which satisfies (2). The equilibrium of (2) is the solution of the equation
Definition 9 (see [20]). Let be fuzzy numbers with , Then the metric of and is defined asand is a complete metric space.
Definition 10 (see [23]). 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 11. Let be a positive equilibrium of (2). The positive equilibrium is stable, if for every , there exists a such that for every positive solution of (2), which satisfies , we have for all The positive equilibrium is asymptotically stable, if it is stable and every positive solution of (2) converges to the positive equilibrium of (2) with respect to as
3. Main Results
3.1. Existence and Uniqueness of Positive Solution
First we study the existence and uniqueness of the positive solutions of (2). We need the following lemma.
Lemma 12 (see [33]). Let be continuous; are fuzzy numbers. Then
Theorem 13. Consider (2) where are positive fuzzy numbers, if there exists fuzzy number such that for Then for any positive fuzzy numbers , there exists a unique positive solution of (2).
Proof. The proof is similar to heorem 2.3 [24]. Suppose that there exists a sequence of fuzzy numbers satisfying (2) with initial conditions . Consider -cuts, , and applying Lemma 12, and for , we haveIt follows from (12) and H-difference of fuzzy numbers that, for ,Then it is obvious that, for any initial condition , there exists a unique solution
Now we prove that , where is the solution of system (13) with initial conditions , determines the solution of (2) with initial value , such that,From Definition 2 and since are positive fuzzy numbers, for any , we haveWe claim that, for ,We prove it by induction. It is clear that (16) is true for . Suppose that (16) holds true for . Then, from (13), (15), and (16) for , it follows that Therefore (16) is satisfied. Moreover, for , it follows from (13) thatSince are positive fuzzy numbers, from Definition 2, then , , , , , are left continuous. From (18) we have , 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 , and since and are positive fuzzy numbers, there exist constants , , , , , such that, for ,Hence, from (18) and (19), for , we getfrom which it is clear thatTherefore (21) implies is compact, and Deducing inductively it can follow easily thatTherefore, from (16), (22) and , are left continuous. It can conclude that determines a sequence of positive fuzzy numbers such that (14) holds.
We prove that is a solution of (12) with initial value . Since , Namely, is a solution of (12) with initial value .
Suppose that there exists another solution of (12) with initial value . Then from arguing as above we can easily get that, for ,Then from (14) and (24), we have , and hence , and this completes the proof of Theorem 13.
3.2. Dynamical Behaviour of Positive Solution
In order to study the dynamical behavior of the solution to (2), we first consider the following system of difference equationsIt is clear that the equilibrium points of (25) include the following four cases:
Lemma 14. Consider difference equationIf , then .
Proof. Consider function , and we have . It follows thatIt is clear that the maximum of is equal to Therefore we have
Lemma 15 (see [2]). Suppose the vector difference equationwhere . Let be an equilibrium point (29), and denote the Jacobian Matrix of function at . Then
(i) is called a hyperbolic equilibrium if has no eigenvalues with absolute value equal to 1.
(ii) is called a sink or an attracting equilibrium if every eigenvalue of has absolute value less than 1.
(iii) is called a source or a repelling equilibrium if every eigenvalue of has absolute value greater than 1.
(iv) is called a saddle point if some of the eigenvalues of are greater and some are less than 1 in absolute value.
Theorem 16. Consider the system of difference equations (25), where , and are positive real constants, and the initial values are positive real numbers; then the following statements are true.
(i) The equilibrium is a source (a repelling equilibrium).
(ii) The equilibrium is a saddle point if ; the equilibrium is a source (a repelling equilibrium) if .
(iii) The equilibrium is a saddle point if ; the equilibrium is a source (a repelling equilibrium) if .
Proof. (i) It is clear that is always an equilibrium. We can easily obtain that the linearized system of (25) about the positive equilibrium iswhereWe can easily obtain that the eigenvalue of Jacobian matrix , , . It is clear that . This implies that the equilibrium is a source (a repelling equilibrium).
(ii) We can obtain that the linearized system of (25) about the equilibrium iswhereIt is easy to obtain the following: if , the eigenvalue of , , , then is a saddle point. if , all the eigenvalue of , , then is a source (a repelling equilibrium).
(iii) The proof of (iii) is similar to the proof of (ii). So we omit it.
Theorem 17. Consider the system of difference equations (25), where the initial values are positive real numbers. If , then (25) has a unique positive equilibrium point which is globally asymptotically stable.
Proof. It follows easily from (25) that is a unique positive equilibrium point.
We can obtain that the linearized system of (25) about the positive equilibrium iswhereWe can easily obtain that the eigenvalue of Jacobian matrix , . It is clear that . Hence the equilibrium is a sink (an attracting equilibrium).
Noting Lemma 14, we havewhere . Then from (25) we haveFrom this, we haveThe relation (38) implies , . We claim thatSuppose on the contrary that From (38), we havefrom which we haveIt follows from (41) that and , which is contradicting with . So . Hence we have , Therefore the unique positive equilibrium is globally asymptotically stable.
Theorem 18. Consider the fuzzy difference equation (2), where are positive fuzzy numbers. Consider that there exists positive constants , for all such thatThen the following statements are true.
(i) Every positive solution of (2) is bounded.
(ii) Equation (2) has a unique positive equilibrium .
(iii) Every positive solution of (2) converges the unique equilibrium with respect to as
Moreover the unique positive equilibrium is asymptotically stable.
Proof. (i) Let be a positive solution of (2) with initial conditions . Suppose that (19) holds; from (25) and using Lemma 14, we get thatFrom (19) and (43), we have that for all where . From (44), we have for , , so . Thus the proof of Part (i) is completed.
(ii) We consider the systemThen the positive solution of (45) is given byLet be a positive solution of (2) such that , , Then applying Theorem 17 to system (13) we haveFrom (44) and (47) we have for ,Since , , , are left continuous, from (45), it follows that are also left continuous. From (46) and (19), we haveTherefore (49) implies that , and so It is clear thatSo from Definition 2, (45), (48), (50) and determine a fuzzy number such thatProvided that there exists another positive equilibrium of (2), then there exist functions such thatFrom this we getSo . Hence . This completes the proof of Part (ii).
(iii) From (47) we haveNamely, every positive solution of (2) converges the unique equilibrium with respect to as
Let be an arbitrary positive real number; we consider the positive real number as followswhere .
Let be a positive solution of (2) such that . From this we haveFrom (18), (19), and (45), we havewhere , ,
From (55), it is obvious thatFrom this and working inductively we can easily prove thatAnd so Therefore the positive equilibrium is stable. Moreover relation (54) holds. So the equilibrium is asymptotically stable. The proof of Theorem 18 is completed.
Remark 19. From Theorem 18, we can know that the dynamical behavior of (2) is relevant to the intrinsic growth rate of population. No matter what the initial population quantity is small and no matter what the carrying capacity of the surround environment is large. As long as the intrinsic growth rate of population satisfies , for all , the solution of fuzzy difference equation (2) is bounded and eventually converges to the unique positive equilibrium.
4. An Illustrative Example
In order to illustrate our obtained results, we give a numerical example to show effectiveness of theoretic results.
Example 1. Consider discrete time fuzzy single species modelwhere and the initial value are positive triangular fuzzy numbers such thatfrom which, we get, for ,From (61) and (63), it results in a coupled system of difference equation with parameter ,It is clear that (42) of Theorem 18 is satisfied. Therefore the solution of (61) converges to the unique positive equilibrium (see Figures 1–5).
5. Conclusion
Difference equation is one of the most important models when it is applicable in various problems in different fields. It is also important if it is applied to study population dynamic in fuzzy environment. When it can be studied in fuzzy environment the behavior of it changes. In this work, we first consider the dynamical behavior of discrete time single species with fuzzy parameters and fuzzy initial conditions. Compared with crisp discrete time single species model, the dynamical behavior of system is different. The parameters of system and satisfy the condition , for all , and the solution of (2) is bounded and converges to unique fuzzy positive equilibrium.
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)), by the Priority Projects of Science Foundation at Guizhou University of Finance and Economics (No. 2018XZD02), by the Innovative Exploration and Academic Planted Talent Project of the Guizhou University of Finance and Economics, and by the Scientific Research Foundation of Guizhou Science and Technology Department 1051).