Abstract
A single species population model is investigated, where the discrete maturation delay and the Ricker birth function are incorporated. The threshold determining the global stability of the trivial equilibrium and the existence of the positive equilibrium is obtained. The necessary and sufficient conditions ensuring the local asymptotical stability of the positive equilibrium are given by applying the Pontryagin's method. The effect of all the parameter values on the local stability of the positive equilibrium is analyzed. The obtained results show the existence of stability switch and provide a method of computing maturation times at which the stability switch occurs. Numerical simulations illustrate that chaos may occur for the model, and the associated parameter bifurcation diagrams are given for certain values of the parameters.
1. Introduction
The model of a single species population growth is usually the base of modeling transmission of some infection and interaction between two or more species. Cooke et al. [1] proposed the model of a single species population model Here denotes the mature population of the species; and are the death rates of the immature and mature population, respectively; the delay is the maturation time, and is the probability in which an immature individual keeps surviving to mature; the Ricker function represents the per capita birth rate of mature individuals, which reflects the dependence of population birth on the density of individuals, and the parameter is the per capita maximal birth rate. Model (1) has been applied to describe some epidemiological and population biological models [1–5].
For model (1), Cooke et al. [1] found the existence of stability switch as increases, by using the geometric method. Jiang and Zhang [6] theoretically discussed the stability switch of model (1) by means of the geometric criterion proposed by Beretta and Kuang [7], where is used as the bifurcation parameter. Wei and Zou [8] considered the local and global Hopf bifurcation for (1), where is used as the bifurcation parameter.
In this paper, our aim is to investigate the stability of model (1) and completely analyze the effect of all the parameter values on the stability. By making the suitable scaling, model (1) is reduced, and the necessary and sufficient conditions for the stability of the positive equilibrium of the simplified model are obtained. The stability switch is also proved theoretically, and the associated conditions are given. The obtained results supplement the conclusions in [1, 6, 8], and numerical simulations illustrate the existence of chaos for certain parameter values. The associated parameter bifurcation diagrams are plotted for certain values of the parameters.
The paper is organized as follows. In the next section, model (1) is reduced, and the stability is analyzed by the LaSalle’s invariance principle and the Pontryagin’s method. In Section 3, the effect of all the parameter values on the stability of the positive equilibrium is discussed, and the associated parameter bifurcation diagrams are given for certain values of the parameters to show the complexity of dynamical behaviors of the model. Finally, a brief conclusion is given.
2. Stability Analysis
For (1), making the scaling and denoting , then dropping their bars yields
Denote still by , then (3) becomes
From the biological meaning, the initial conditions for (4) are given as follows: where and .
The following theorem establishes the positivity and boundedness of solutions of (4).
Theorem 1. All solutions of (4) under the initial condition (5) are positive on and ultimately bounded.
Proof. Assume that there is () such that ; then it follows from and the continuity of solution of (4) that there is such that for . So we have . However, . This contradiction implies that for . Therefore, all solutions of (4) under the initial condition (5) are positive on .
On the other hand, we know that for ; then, under the initial condition (5), from (4) we have . It follows that ; that is, all solutions of (4) under the initial condition (5) are ultimately bounded, and the set is positively invariant for (4).
The proof of Theorem 1 is complete.
Obviously, (4) always has the trivial equilibrium , and, when , that is, , (4) also has a unique positive equilibrium , where since . Notice that implies that .
With respect to the stability of the trivial equilibrium , we have the following statement.
Theorem 2. The trivial equilibrium of (4) is globally stable as and unstable as .
Proof. Define a Lyapunov functional , then the derivative of with respect to along solutions of (4) is given by
For ,
When , and the equality holds if and only if . It is easy to know that the largest invariant set of (4) on the set is the singleton . It follows by the LaSalle’s invariance principle [9] that the trivial equilibrium of (4) is globally stable on the set as .
The characteristic equation of (4) at is given by
Since for , and , equation must have positive root as . Therefore, the trivial equilibrium of (4) is unstable as .
The proof of Theorem 2 is complete.
Remark 3. From biological meaning, the global stability of the trivial equilibrium implies the eventual extinction of the population, and its instability implies the persistence of the population. Therefore, the population is extinct finally if and keeps survival if .
In the following we consider the stability of the positive equilibrium . The linearized equation of (4) at is given by Substituting with into (9), we get the characteristic equation
Obviously, the root of (10) with is for ; that is, the positive equilibrium is locally asymptotically stable as and . Therefore, for , with variation of stability of the positive equilibrium can change only when the pure imaginary roots of (10) appear.
For the stability of , analysis is realized by considering two cases: and . For the case , we have the following statement.
Theorem 4. The positive equilibrium of (4) is locally asymptotically stable if .
Proof. Suppose that there is a root of (10) with nonnegative real part if , then denote the root by , where . Substituting it into (10) gives
Note that is equivalent to ; then, for , the norm of the left-hand side of (11) is not less than one, but the norm of the right-hand side is less than one. So the contradiction occurs; that is, the assumption does not hold. Therefore, all roots of (10) are with negative real parts as .
When , (10) becomes
Note that and any pure imaginary number are not the root of (12) and that (12) with only has the root ; then all roots of (12) are with negative real parts.
Summarizing the above inferences, the positive equilibrium is locally asymptotically stable as .
This completes the proof of Theorem 4.
In the following we consider the case ; that is, , by applying the Pontryagin’s method [10], which is introduced in the appendix.
Let , then (10) can become Separating the real and imaginary parts of gives where
According to the Pontryagin’s method, we first discuss zeros of and then consider position of the roots of (13) on the complex plane. With respect to zeros of we have the following statement.
Proposition 5. All zeros of are real.
Proof. If , then and the function in Theorem 10 in Appendix is . Therefore, we may take in Theorem 10. From Theorem 10, all zeros of are real if and only if there are real zeros of in the interval for a sufficiently large integer. Observe that and that is not zero of ; then, for , is equivalent to the equation
According to the graphics of functions and (Figure 1), it is easy to see that (16) has roots in the interval , denoted by , , and . Since both functions and are odd functions, (16) also has roots in the interval , denoted by , , and . Thus, together with the zero of , , has exactly zeros in the interval for any positive integer. It follows from Theorem 10 that Proposition 5 is true.
Theorem 9 in the appendix gives the necessary and sufficient conditions ensuring that all roots of (13) have negative real parts. So, by applying Theorem 9, we may get the necessary and sufficient conditions ensuring that is locally asymptotically stable. The corresponding result is as follows.
Theorem 6. When , the positive equilibrium of (4) is locally asymptotically stable if and only if , where is the zero of function in the interval .
Proof. Proposition 5 has shown that all zeros of are real, then according to Theorem 9, we only need to verify that holds for all zeros of .
Since
for each root of (16) we have
Then, according to the position of zeros of , we have for , and for .
Note that the inequality is equivalent to the inequality , then . So for . Therefor we have
From we get . Substituting it into gives
Since and is an even function of , we will only prove that for , as .
Let , then , and it is easy to see that is increasing with increasing of . And
By monotonicity of with respect to , is increasing as increases. So implies that ; that is, . Since , we get for , when .
The proof of Theorem 6 is complete.
3. Dependence of Stability of Equilibria on the Values of Parameters
In the previous section, we have analyzed the stability of equilibria of (4). In this section, we will investigate the dependence of stability of equilibria on the values of all parameters.
For an arbitrary positive number , (16) has a unique root in the interval . This root is only determined by the value of and is independent of the parameters and . Then, for a given positive number , the conditions in Theorem 6 can be expressed in the inequalities . Combining the condition in Theorem 4, the positive equilibrium of (4) is locally asymptotically stable if and only if for and and unstable if for and . Further, according to Theorems 2, 4, and 6, for an arbitrary given positive number , we can determine the stability of equilibria and of (4) in the associated region in the plane (Figure 2). That is, denote then, for the given , when , is globally stable; when , is locally asymptotically stable, and is unstable; when , both and are unstable. Biologically, the population is extinct eventually as and persistent as .
According to Figure 2, it is obvious that the range of , in which the population is extinct, is enlarging as increases.
From Figure 2, for any given and , when increases from zero, (4) first has no positive equilibrium and its trivial equilibrium is globally stable; when passes through the curve , the trivial equilibrium is unstable, and the positive equilibrium appears and it is locally asymptotically stable; when it passes through the curve again, both the trivial and the positive equilibria are unstable. For any given , if , the trivial equilibrium is globally stable for an arbitrary ; if , the locally asymptotically stable positive equilibrium could disappear as increases and passes through the curve ; if , the stability of the positive equilibrium could change from unstable to stable, then to disappearing as increases and passes through the curves and .
On the other hand, for the given and , how does the value of affect the stability of equilibria and ?
From Theorems 2 and 4, is globally stable if and only if ; is locally asymptotically stable if . Then, how is the situation ?
We define function for . Straightforward computation yields for , where both and are continuous in the interval .
Note that then function has a unique extreme point in the interval , denoted by , and its maximum is .
We further discuss function for , where is the root of equation in the interval for . Here, . Therefore, with respect to sign of function in , we have the following statements. If , for ; If and , for ; If and , has exactly two zeros in the interval , and (), such that for and for .
For function defined by (16) for , it is continuous and increasing in the interval , since it follows from (16) that for . And it is easy to know that and . Further, according to the properties of function and the definition of , for the given and , changes from to as increases from to . Since for , the signs of functions and are opposite. Notice that and satisfy (16), then function has the opposite sign to function . Denoting the values of corresponding to and by and , respectively, from Theorems 4 and 6 we have the following results.
Theorem 7. The positive equilibrium of (4) is locally asymptotically stable if one of the following conditions is satisfied:(i);(ii) and ;(iii), , and ;(iv), , and ;(v), , and .
The positive equilibrium of (4) is unstable if , and .
Additionally, the condition for the existence of the positive equilibrium is , so we can also give the other kinds of expressions with respect to Theorem 7 in the following.
Theorem 8. Only if the positive equilibrium of (4) exists, it is locally asymptotically stable when one of the following conditions is satisfied:(i);(ii) and ;(iii), , and .
When the positive equilibrium of (4) exists, with increasing of its stability can change from stable to unstable and to stable again when the conditions: , and are satisfied. And the change occurs in turn as passes through and .
Theorem 7 or 8 shows that the stability of the positive equilibrium of (4) does not change with variation of value of as one of the first three conditions in them is satisfied and that can undergo the stability switch with increasing of as and . And the stability switch happens at and . Correspondingly, it can be verified that the Hopf bifurcation also occurs at and . Numerical simulations illustrate the existence of periodic solution and chaos as (Figure 3).
(a)
(b)
(c)
(d)
On the other hand, when the parameters , and are used as the bifurcation parameter respectively, the associated parameter bifurcation diagrams are given in Figures 4, 5, and 6, respectively. They may show the complexity of dynamic behaviors of model (5), including chaos.
(a)
(b)
(c)
4. Conclusion
In this paper, we first proved the positivity and the ultimate boundedness of model (1) and obtained the threshold determining the global stability of the trivial equilibrium and the existence of the positive equilibrium. Next, the stability of the positive equilibrium is investigated by means of Pontryagin’s method, and the necessary and sufficient conditions ensuring the local stability of the positive equilibrium are obtained. Lastly, the dependence of stability of the positive equilibrium on the parameter values is analyzed, and the stability switch with variation of the maturation time is discussed completely. Additionally, numerical simulations exhibit that chaos may occur for certain parameter values and show that the local asymptotical stability of the positive equilibrium implies its global stability. These results of numerical simulations need to be further proved rigorously. On the other hand, in order to show the dynamic complexity of (1), we also gave the associated parameter bifurcation diagram.
Appendix
We here introduce a method for determining when the roots of characteristic equation are in the left half-plane, where is a polynomial in . The method is due to Pontryagin [10], which is referred to as the Pontryagin’s method.
Suppose that is a polynomial in , where is called the principal term of the polynomial if and, if, for each other term with , we have either , , or . Clearly, not every polynomial has a principal term.
Theorem 9. Let , where is a polynomial with principal term. Suppose that , is separated into its real and imaginary parts, . If all zeros of have negative real parts, then the zeros of and are real, simple, and alternate and for . Conversely, all zeros of will be in the left half-plane provided that either of the following conditions is satisfied.(i)All the zeros of and are real, simple, and alternate and inequality (A.2) is satisfied for at least one .(ii)All the zeros of are real and, for each zero, inequality (A.2) is satisfied.(iii)All the zeros of are real and, for each zero, inequality (A.2) is satisfied.
On the other hand, in order to prove that all the zeros of or are real, we need the following results. Suppose that is a polynomial in with real coefficients which has the form where is a homogeneous polynomial of degree in . The principal term in the polynomial is defined as the term for which either , ,, , or for all other terms in (A.3).
Let denote the principal term of in (A.3); let denote the coefficient of in , and let
Theorem 10. Let be a polynomial with principal term . If is such that , , then, for sufficiently large integers , the function will have exactly zeros in the strip . Conversely, the function will have only real roots if and only if, for sufficiently large integer , it has exactly roots in the strip .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Fund of China (nos. 11371369 and 11371313).