Stability Analysis of a Population Model with Maturation Delay and Ricker Birth Function

and Applied Analysis 3 than one. So the contradiction occurs; that is, the assumption does not hold. Therefore, all roots of (10) are with negative real parts as 0 < ln b − d 1 τ < 2. When ln b − d 1 τ = 2, (10) becomes λ + 1 + e λτ = 0. (12) Note that λ = 0 and any pure imaginary number are not the root of (12) and that (12) with τ = 0 only has the root λ = −1; then all roots of (12) are with negative real parts. Summarizing the above inferences, the positive equilibrium N∗ is locally asymptotically stable as 0 < ln b − d 1 τ ≤ 2. This completes the proof of Theorem 4. In the following we consider the case ln b − d 1 τ > 2; that is, 1 + d 1 τ − ln b < −1, by applying the Pontryagin’s method [10], which is introduced in the appendix. Let ρ = λτ, then (10) can become P (ρ, e ρ ) := ρe ρ + τe ρ − τ (1 + d 1 τ − ln b) = 0. (13) Separating the real and imaginary parts of P(iω, eiω) gives P (iω, e iω ) = F (ω) + iG (ω) , (14) where F (ω) = − ω sinω + τ [cosω − (1 + d 1 τ − ln b)] , G (ω) = ω cosω + τ sinω. (15) According to the Pontryagin’s method, we first discuss zeros of G(ω) and then consider position of the roots of (13) on the complex plane. With respect to zeros of G(ω) we have the following statement. Proposition 5. All zeros of G(ω) are real. Proof. If g(ω, u, V) = ωu + τV, then G(ω) = g(ω, cosω, sinω) and the function Φ(s) ∗ (ω) in Theorem 10 in the appendix is cosω. Therefore, we may take ε = 0 in Theorem 10. From Theorem 10, all zeros of G(ω) are real if and only if there are 4k + 1 real zeros of G(ω) in the interval [−2kπ, 2kπ] for k a sufficiently large integer. Observe that G(0) = 0 and that ω = nπ(n = ±1, ±2, . . .) is not zero of G(ω); then, for ω ̸ = 0, G(ω) = 0 is equivalent to the equation − τ ω = cotω. (16) According to the graphics of functions −τ/ω and tan ω (Figure 1), it is easy to see that (16) has 2k roots in the interval (0, 2kπ), denoted by ω n , n = 1, 2, 3, . . . , 2k, and ω n ∈ ((n − 1/2)π, nπ). Since both functions −γ/ω and tan ω are odd functions, (16) also has 2k roots in the interval (2kπ, 0), denoted by ω −n , n = 1, 2, 3, . . . , 2k, and ω


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;  1 and  are the death rates of the immature and mature population, respectively; the delay  is the maturation time, and  − 1  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][2][3][4][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.
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 [0, +∞) and ultimately bounded.
The proof of Theorem 1 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 In the following we consider the stability of the positive equilibrium  =  * .The linearized equation of ( 4) at  =  * is given by Substituting  =   with  ̸ = 0 into (9), we get the characteristic equation Obviously, the root of (10) with  = 0 is  = − ln  < 0 for  > 1; that is, the positive equilibrium is locally asymptotically stable as  = 0 and  > 1.Therefore, for  > 1, with variation of  stability of the positive equilibrium  * can change only when the pure imaginary roots of (10) appear.
For the stability of N = N * , analysis is realized by considering two cases: 0 < lnb−d 1  ≤ 2 and lnb−d 1  > 2. For the case 0 < lnb − d 1  ≤ 2, we have the following statement.

Theorem 4. The positive equilibrium 𝑁
Proof.Suppose that there is a root of (10) with nonnegative real part if 0 < ln  −  1  < 2, then denote the root by  =  + , where  ≥ 0. Substituting it into (10) gives Note that 0 < ln − 1  < 2 is equivalent to |1+ 1 −ln | < 1; then, for  ≥ 0, 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 0 < ln  −  1  < 2.
When ln  −  1  = 2, (10) becomes Note that  = 0 and any pure imaginary number are not the root of (12) and that (12) with  = 0 only has the root  = −1; then all roots of (12) are with negative real parts.Summarizing the above inferences, the positive equilibrium  * is locally asymptotically stable as 0 < ln  −  1  ≤ 2.
This completes the proof of Theorem 4.
In the following we consider the case ln  −  1  > 2; that is, 1 +  1  − ln  < −1, 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.

𝐹 (𝜔
The proof of Theorem 6 is complete.

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.
According to Figure 2, it is obvious that the range of  1 , in which the population is extinct, is enlarging as  increases.
From Figure 2, for any given  1 and , when  increases from zero, (4) first has no positive equilibrium and its trivial equilibrium is globally stable; when  passes through the curve  1 , the trivial equilibrium is unstable, and the positive equilibrium appears and it is locally asymptotically stable; when it passes through the curve  2 again, both the trivial and the positive equilibria are unstable.For any given , if 0 <  ≤ 1, the trivial equilibrium is globally stable for an arbitrary  1 ; if 1 <  ≤ exp(1 − sec  1 ), the locally asymptotically stable positive equilibrium could disappear as  1 increases and passes through the curve  1 ; if  > exp(1 − sec  1 ), the stability of the positive equilibrium could change from unstable to stable, then to disappearing as  1 increases and passes through the curves  2 and  1 .
On the other hand, for the given  and  1 , how does the value of  affect the stability of equilibria  = 0 and  =  * ?
For function  = () ∈ (/2, ) defined by ( 16) for  ∈ (0, +∞), it is continuous and increasing in the interval (0, +∞), since it follows from (16) that for  ∈ (/2, ).And it is easy to know that lim  → 0 Additionally, the condition for the existence of the positive equilibrium  =  * is  < ln / 1 , 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)  ≤  2 ; (ii)  >  2 and  ≤  * ; (iii)  >  2 ,  >  * , and ( * ) < 0. 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:  >  2 ,  >  * , and ( * ) > 0 are satisfied.And the change occurs in turn as  passes through  * 1 and  * 2 .
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 ( * ) > 0. And the stability switch happens at  =  * 1 and  =  * 2 .Correspondingly, it can be verified that the Hopf bifurcation also occurs at  =  * 1 and  =  * 2 .Numerical simulations illustrate the existence of periodic solution and chaos as  ∈ ( * 1 ,  * 2 ) (Figure 3).On the other hand, when the parameters ,  1 , 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.

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.
Theorem 7. The positive equilibrium  =  * of (4) is locally asymptotically stable if one of the following conditions is satisfied: