Stability analysis for a new fractional order N species network

Abstract: The present paper considers a fractional-order N species network, in which, the general functions are used for finding general theories. The existence, uniqueness, and non-negativity of the solutions for the considered model are proved. Moreover, the local and global asymptotic stability of the equilibrium point are studied by using eigenvalue method and Lyapunov direct method. Finally, some simple examples and numerical simulations are provided to demonstrate the theoretical results.


Introduction
Population dynamics is one of the main issues of theoretical biomathematics and has aroused the interest of many scholars in the fields of mathematical biology. A lot of methods have been presented and developed to study the evolution of population dynamics. It is clear that establishing mathematical models is the prerequisite for analysis of population dynamics. Until now, researchers have proposed a variety of models, represented by difference equations or differential equations [1][2][3].
As is well known, no species can exist in isolation, all populations must be related to the others directly or indirectly, and mutualism, parasitism, competition and predation are the four most widespread direct relationships in the ecosystem [4]. Based on these four relationships, researchers have established a variety of models to study the dynamics of the population, such as the Lotka-Volterra models, Leslie-Gower models and predator-prey models [5][6][7][8]. In recent years, many researchers began to study more complex and realistic population models and have got some interesting results [9][10][11][12][13][14]. In [15], a predator-prey system with Beddington-DeAngelis functional response was considered, and are drawn in Section 5.

Preliminaries
In this section, some definitions and lemmas in fractional calculus are recalled, which will be used later. Definition 1. [46] The Caputo fractional derivative of a function f (t) with order α is defined as where α > 0, t 0 is the initial time, n ∈ N, and Γ is the Gamma function.
Lemma 1. [47] The equilibrium points x * of D α x(t) = f (x) are locally asymptotically stable if all eigenvalues λ i of the Jacobian matrix J = ∂ f /∂x evaluated at the equilibrium points satisfy | arg(λ i )| > απ 2 . Lemma 2.
x) and satisfies the conditions of Lemma 3, then x = x * is uniformly asymptotically stable.

Model description
In [50], the following model is considered.
where the functions a i (t), b i j (t) are assumed to be nonnegative and continuous for t ∈ (−∞, ∞). This system is commonly called the Volterra-Lotka system for N-competing species. Because of the complex relations between the species, it would be more realistic to change the functions b i j (t)u j (t) to f i j (u j (t)), so one can build a new population model. This paper considers the following generalized fractional-order ecosystem model: where t 0 ≥ 0 is the initial time, x i (t) for i = 1, 2, 3, . . . , N denotes population density of the species x i at time t, a i j = ±1, 0. f i j (x j (t)) are continuous, non-negative and differentiable functions. f ii (x i ) are the specific growth rate of x i in the absence of other species for i = 1, 2, 3, . . . , N. If x i represents a producer in ecosystems, one can see that the specific growth rate of x i is positive, so one makes a ii > 0(= 1). If x i represents a consumer in ecosystems, one can see that the specific growth rate of x i is negative, so one makes a ii < 0(= −1). The greater the number of species is, the greater the intra-specific competition will be. So d f ii (x i )/dx i < 0, if x i represents a producer and d f ii (x i )/dx i ≥ 0, if x i represents a consumer.
f i j (x j ) are the effect of x j on x i in a time unit for i = 1, 2, 3, . . . , N, i j. The greater x j is the greater the impact on x i will be, so one makes d f i j (x j )/dx j > 0. It is easy to see that when x j = 0, there is no effect on x i . That is f i j (x j ) = 0 for x j = 0.
One can describe the relationship between x i and x j by determining the sign of a i j and a ji . If a i j > 0(= 1) and a ji > 0(= 1), then the relationship between x i and x j is mutualism. If a i j > 0(= 1) and a ji < 0(= −1), then the relationship between x i and x j is predation(parasitism) and it can be seen that x i is the predator(parasite) and x j is the prey (host). If a i j < 0(= −1) and a ji < 0(= −1), then the relationship between x i and x j is competition. If a i j = 0 and a ji = 0, there is no direct relationship between x i and x j .
One should consider the significance of system (6), that is the boundedness of x i . So the function f i j (x j (t)) for i, j = 1, 2, 3, . . . , n should give the restriction of making x i to be bounded. Remark 2. If x i represents a producer, since d f ii (x i )/dx i < 0 and f ii (x i ) > 0, then f ii (x i ) = b + g(x i ) (g(x i ) does not contain a constant term and b is a constant), where b > 0 represents growth and dg(x i )/dx i < 0. In addition, if the time dependence of coefficients is ignored, Eq (5) can be regarded as a special form of Eq (6) (just set all x i as a producer, a ii = 1, a i j = −1, f i j (x j ) = b i j x j and α = 1). (6) is same as the model in [32]. Let x 1 as a producer and x 2 as a consumer, a 11 , a 21 = 1, a 12 , a 22 = −1, f 11 = r − kx 1 , f 12 = x 2 , f 21 = βx 1 and f 22 = 1, then Eq (6) is same as system (5) in [33]. Let x 1 as a producer and x 2 , x 3 as consumers, a i1 = 1, 23 x 3 and f 32 = c 32 x 2 , then Eq (6) is same as system (1) in [34]. Just let a 23 = a 32 = 0, then Eq (6) is same as system (19) in [34]. So the models in [32][33][34] are special forms of Eq (6).

Existence and uniqueness of the solution
Theorem 1. If x i (t 0 ) > 0, for i = 1, · · · , N, then is a unique solution x(t) of system (6) Proof. We study the existence and uniqueness of the solution of system (6) For any X,X ∈ Ω Obviously, H(Z) satisfies the Lipschitz condition. Thus, system (6) has a unique solution x(t).
Proof. Let x i (t 0 ) > 0, i = 1, · · · , N in R N + be the initial solution of system (6). By contradiction, suppose that there exists a solution x(t) that lies outside of R N + . The consequence is that x(t) crosses the x i axis for i = 1, 2, 3, . . . , N.
Assume that x(t) crosses x k = 0, then there exists t * such that t * > t 0 and x k (t * ) = 0, and there exists t 1 sufficiently close to t * such that t 1 > t * , and x k (t) < 0 for all t ∈ (t * , t 1 ]. Based on the k-th equation of system (6), one obtains Since x(t) and f k j (x j (t)) are continuous, one can see that By using the Laplace transform on both sides, one gets where E α is the Mittag-Leffler function. Thus, x(t) ≥ 0 for any t ≥ t 0 , which contradicts the assumption. By using the same method, one gets x i (t) are non-negative for i = 1, 2, . . . , N.

Stability analysis of equilibrium points
In this section, the stability of the equilibrium points will be investigated by using the Lyapunov direct method. The equilibrium points of the model can be obtained from the following equations.
In the following, we will only consider the stability of a certain equilibrium point, for the other equilibrium points can be discussed by the same way.
Remark 4. The conditions of Theorem 3 are sufficient conditions. If H j < 0 and all eigenvalues of λ k + a 1 λ k−1 + a 2 λ k−2 + · · · + a k−1 λ + a k = 0 satisfy the condition in Lemma 1, the equilibrium point is also locally asymptotically stable. 0) is the equilibrium point of system (6), then x * is globally asymptotically stable, if the following algebraic conditions hold: 1). x * satisfy k j=1 a i j f i j (x * j ) + a ii f ii (x * j ) ≤ 0, for i = k + 1, k + 2, . . . , N. 2). ∃k i j ≥ 0 satisfy n j=1 k i j = 1 for i, j = 1, 2, . . . , N and i j. 3). ∃c i > 0 and ∃k i j ≥ 0 satisfy 2 a ii a j j c i c j k i j k ji f ii (

Global stability
Let us consider the following positive definite function about x * One can subtly choose c i > 0 such that V is a Lyapunov function. By Lemma 2, one obtains For arbitrary ε > 0, since functions f i j for i, j = 1, 2, . . . , n are differentiable functions, there exists Thus By the arbitrariness of ε, one has For i j, then one has According to the condition 2 a ii a j j c i c j k i j k ji f ii ( Thus, x * is a globally asymptotically stable equilibrium point for the system (6).
Remark 5. According to theorem 4, one can see that N-order system should have N(N−1)/2 conditions at least. So if N is large enough, it is difficult to find the suitable c i , k i j for i, j = 1, 2, 3, . . . , n and i j.

Numerical simulation
In this section, we will give some examples to demonstrate the theoretical analysis.

Example 3:
The new example will consider the complex relationships between the species, one selects the functions of the system (6) as follow.

Conclusions
In this paper, one considers a class of generalized fractional population model. Firstly, the existence, uniqueness and non-negative of solutions of the system are proved. Secondly, one provides a sufficient conditions for local stability and globally stability of the equilibrium points. Finally, three examples are given to demonstrate the theoretical analysis. This paper not only extends the integer population models to fractional-order form, but also extends the dimension to n-dimension. The new model can be used to simulate the relationships between various populations in a complex ecosystem. Compared with other population models, our model is more general.
Time delay stays in the biological system widely and may effect the stability of the biological system. Thus, system (6) with time delay is of great research value. It will be our future work.