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Continuous Time Multi Stage Population Model discrete

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1. Introduction

Several researchers have focused in recent years on the dynamics of nonautonomous discrete-time models and the advantage of seasonal versus continuous breeding in terms of maximizing the total population number or the total number of adults over a fixed time period 1–10. These studies were motivated by an experimental system which investigated the responses of populations of the flour beetle, Tribolium castaneum, cultured in a series of regularly fluctuating environments. Therein, it was observed that population density declined as environmental period lengthened 11.

Motivated by an urban population of green treefrogs that we are studying 12, we recently developed the following juvenile–adult model for a seasonally breeding population 13: where the parameters a, c, k 1 and k 2 are assumed to be positive. Moreover, since represents the fraction of the juveniles x t present at time t that survive one unit of time and appear as adults at time t + 1, we assumed a > 1 for the model to be biologically meaningful (note that 1/a is the inherent survivorship of juveniles). Similarly, we assume that the parameter c > 1. The function b t represents the time-dependent birth rate.

In that paper we focused on the following question: given that an adult recruits a fixed number of juveniles in one year, is it advantageous (in terms of maximizing the total number of adults over a period of one year) to reproduce continuously or seasonally? To answer this question, we investigated the model dynamics for two types of recruitment: continuous (i.e. b t = b > 0 for t = 0, 1, 2, …) and periodic with period two (i.e. , b 1 = 0, ).

Our analysis showed that for low birth rates, the population which produces seasonally may survive while the one which reproduces continuously will go to extinction. Thus seasonal breeding is beneficial for such values of birth rates. Furthermore, we show that for low values of birth rates where both populations persist, the adults for the continuously breeding population have a lower average over a one-year period than the one that produces seasonally. Therefore, seasonal reproduction is beneficial in this case. However, for high birth rates this conclusion reverses, and it is shown that breeding continuously results in higher adult averages over a one-year period.

The purpose of this paper is to continue this investigation for a three-stage discrete time model. In section 2 we present the model and analyze the continuous breeding case, and show that if the inherent net reproductive number is less than unity then the population goes to extinction, while if it is greater than unity then the unique interior equilibrium is globally asymptotically stable. We then analyze the seasonal breeding with period-two and show that if the inherent net reproductive number is less than unity then the population becomes extinct, while if it is greater than unity the unique two-cycle is globally attracting. At the end of this section we compare the two birth types and show that in this case (unlike the two-stage model) breeding seasonally seems always to be deleterious. In section 3 we provide concluding remarks.

2. Model development and analysis

We develop a theoretical model describing the dynamics of a population which engages in seasonal breeding and is divided into three stages: a juvenile stage, nonbreeding (sexually immature) stage, and breeding (adult) stage. To this end, denote by x t the number of juveniles at time t, by y t the number of nonbreeding individuals at time t and by z t the number of adults at time t. We assume that the juvenile and nonbreeding stages are less than or equal to one time unit (i.e. all juveniles and nonbreeders move into the next stage within one time step). Assume that competition occurs within each stage. We then obtain the following nonautonomous three-stage discrete model: where b t is the per-capita birth rate of the breeding adults and s i is the survivorship rate of stage i. We assume that s i satisfies the following assumption:

Such an assumption is satisfied, for example, by a Beverton–Holt type of survivorship function. Before we start analyzing model (1), we recall the following result (see Theorem 1.10 in 14), which will be used in the sequel. Consider a k + 1 order nonlinear difference equation of the form

where and I is an open interval of ℝ.

Theorem 2.1

Let x*∈I be an equilibrium of (2). Suppose F satisfies the following two conditions:

  • (a)F is non-decreasing in each of its arguments, and

  • (b)F satisfies for all .

Then x* is a global attractor of all solutions of (2).

2.1 Continuous breeding

In this subsection we consider model (1) with continuous breeding. In particular, we assume that in model (1) b t ≡ b, a positive constant. Clearly solutions of system (1) remain positive. The system always has a trivial steady state E 0 = (0, 0, 0). The z-component of a nontrivial steady state , must satisfy Notice that the right-hand side of (3) is a strictly decreasing function of z by (H1), with value ba 1 a 2 + a 3 when z = 0, and approaches 0 as z goes to infinity. Therefore, equation (3) has a positive solution if and only if . Consequently, (1) has an interior steady state if and only if where solves (3), , and . The interior steady state is unique whenever it exists. Clearly (4) is equivalent to where R 0 is the inherent net reproductive number. Using Theorem 2.1 the asymptotic dynamics of system (1) can be understood and are summarized below.

Theorem 2.2

If R 0 < 1, then (1) has only the trivial steady state E 0 = (0, 0, 0) which is globally asymptotically stable. If R 0 > 1, then E 0 is unstable and (1) has another equilibrium which is globally asymptotically stable in the interior of .

Proof

Suppose R 0 < 1. Let be a solution of (1). Since and for t ≥ 0, consider the following linear system The eigenvalues λ of the above coefficient matrix A satisfy . It is clear that p 0(1) > 0, p 0(−1) < 0, and by our assumption of R 0 < 1. It follows from the Jury conditions 15 that E 0 is locally asymptotically stable. Since A is nonnegative and irreducible with spectral radius less than unity, we have and hence . As a result, all solutions of (1) converge to E 0 and E 0 is globally asymptotically stable.

Suppose now R 0 > 1. It is clear that E 0 is unstable by the above analysis. We first verify that E 1 is locally asymptotically stable. The linearization of (1) with respect to E 1 yields the following Jacobian matrix J(E 1): where , , and . The eigenvalues λ of J(E 1) satisfy . In the following we shall verify the three Jury conditions 15: p 1(1) > 0, p 1(−1) < 0, and .

Observe that J 21 > 0, J 32 > 0, and J 33 > 0 by our assumptions of (H1). Since satisfies (3), , and , we have by (3) that Substituting the above expression of 1 in yields It is also clear that as J 21 > 0, J 32 > 0, and J 33 > 0. We proceed to verify the last inequality , which is equivalent to . For notational convenience, we rewrite by s 1, by s 2 and by s 3. Replacing 1 by the square of the right-hand side of (3), i.e. then Notice , where , and . Furthermore, . The only negative terms in the above expression are where −bs 1 s 2 s 3 can be cancelled out by one of the 2bs 1 s 2 s 3 in (8) and can be combined with the positive term to obtain . This proved that E 1 is locally asymptotically stable.

To show that E 1 is globally attracting in the interior of , we apply Theorem 2.1. Notice system (1) can be converted into the following third-order scalar difference equation: Since , (9) has a unique positive steady state . It is sufficient to prove that is globally attracting for (9) in (0, ∞). Let Then for all x > 0, y > 0, z > 0. Moreover, for z > 0 and by (3). Hence is globally attracting for (9) in (0, ∞) by Theorem 2.1 and consequently E 1 is globally attracting for (1). This completes the proof. ▪

2.2 Seasonal breeding

In this subsection we assume that breeding is seasonal where the function b t in (1) is periodic with period two. Specifically, we set . Let be given. It is clear that for t > 0. Moreover, x 1 = 0, , , , y 2 = 0 and . Therefore, if (x 0, y 0, z 0) is a part of a two-cycle, then we have x 2 = x 0, y 2 = y 0 = 0 and . As a result, if z 0≠0, then z 0 must satisfy Let Then h(0) = 0, and for z ≥ 0 by (H1). Let H(z) denote the right-hand side of (10) and . Then and for all z ≥ 0 as h ′ > 0 and s i satisfies (H1) for i = 1, 2, 3. We conclude that (10) has a positive solution z* if and only if Condition (11) is equivalent to where is the inherent net reproductive number for the seasonal population. Therefore, (1) has a unique two-cycle if and only if (12) holds, where z* satisfies (10) and

Theorem 2.3

If , then E 0 = (0, 0, 0) is globally asymptotically stable for (1).

Proof

We first show that E 0 is globally attracting by using a simple comparison method. Observe that x 2t + 1 = 0 for t ≥ 0 and y 2t = 0 for t ≥ 1. Also imply i.e. for t ≥ 1. Letting n + i = 2(t + i) for i ≥ 0, consider the following linear system of difference equations: with A 1 = x 2 and B 1 = z 2, i.e. The eigenvalues of the above coefficient matrix satisfy It is clear that , and under the assumption . It follows that . Therefore and as a result, and . We conclude that E 0 is globally attracting. It remains to show that E 0 is locally asymptotically stable.

Since system (1) is periodic with period two, the local stability of E 0 depends on the product of the matrices 16: Denote the resulting product matrix by J 0. Then the eigenvalues of J 0 are 0 and eigenvalues of , where Since and , we see that . Therefore all the eigenvalues of J 0 have modulus less than 1. Hence E 0 is locally asymptotically stable and thus E 0 is globally asymptotically stable. ▪

Suppose now . Then it follows from the proof of Theorem 2.3 that E 0 is unstable. Moreover, (1) has a unique two-cycle: where z* satisfies (10) and . In the following we show that the two-cycle is globally asymptotically stable.

Theorem 2.4

If , then the two-cycle is globally asymptotically stable for system (1) in the interior of .

Proof

We first prove that the two-cycle is locally asymptotically stable. Recall that its stability depends on the eigenvalues of the product of the matrices 16: where and Denote the resulting product matrix by J 1. Then where and The eigenvalues of J 1 consist of 0 and the roots of . For the two-cycle to be locally asymptotically stable, it is necessary and sufficient that |C| < 1−AB < 2. Observe that a 32 > 0, a 33 > 0 by (H1) and we have AB > 0. Thus 1−AB < 2 is trivially true. It remains to verify |C| < 1−AB which reduces to C < 1−AB as C > 0. Using , abbreviate s i (z*) by s i and by , and rewrite z* by x, C < 1−AB is equivalent to The only negative terms in the above expression are and Replacing 1 by the right-hand side of the equilibrium equation (10), we see that the first two negative terms can be canceled out by the terms in the equilibrium equation, while the last negative term can be combined with to yield a positive term Therefore C < 1−AB holds and the two-cycle is locally asymptotically stable.

It remains to show that the two-cycle is globally attracting in the interior of . The proof is similar to the proof of Theorem 2.2. Let be given. Observe that x 2t + 1 = 0 for t ≥ 0 and y 2t = 0 for t ≥ 1. It follows that for t ≥ 1 and Let n + i = 2(t + i) for i ≥ 0. We obtain the following system of first-order difference equations: for n ≥ 1. The system is equivalent to the following second-order scalar equation: Let the right-hand side of (14) be denoted by f(x, y), i.e. and . Then Similarly, Furthermore, (14) has a unique interior steady state z* as . It is clear that It follows from Theorem 2.1 that . Hence and . That is, the even subsequence of the solution converges to (x*, 0, z*) and the odd subsequence converges to . Since is arbitrary, we see that the two-cycle is globally attracting in the interior of . Therefore the two-cycle is globally asymptotically stable for (13) in the interior of . ▪

2.3 Comparison between continuous and seasonal breeding

For the rest of this section we assume that in a continuous or seasonal breeding population an adult reproduces the same number of juveniles in a one-year period. Thus, we let . We are interested to see for what values of b continuous breeding (seasonal breeding) is advantageous in terms of maximizing the average number of adults over a one-year period.

(b) Comparison of the breeding adults for periodic and constant birth rates

Differ-entiate both sides of equilibrium equation (3) with respect to b yields , where and Therefore for . Similarly, letting in (10) and differentiate both sides of (10) with respect to b resulting in , where and with . Therefore for . Since , we conclude that when but sufficiently close to , the constant birth rate has a higher breeding adult equilibrium value, i.e. . Therefore, constant birth rate is more advantageous.

We provide numerical results verifying our theoretical conclusions. In figure 1, we choose a set of parameters that results in the continuous birth population converging to a positive equilibrium while the population with period-two birth rate goes to extinction. In this case, while . Increasing the birth rate, we present the results in figure 2. In this case, R 0 = 1.5 > 1 while . Thus, both populations survive. One converges to a positive equilibrium and the other converges to a positive two-cycle. Clearly, the equilibrium value for adults z t is larger than the average two-cycle. Thus continuous reproduction is advantageous for this birth rate value.

Figure 1. A comparison between continuous and seasonal breeding with period two birth rate for model (1). The survivorship functions are s 1(x) = a 1/(1 + k 1 x), s 2(y) = a 2/(1 + k 2 y), s 3(z) = a 3/(1 + k 3 z) with parameter values a 1 = 0.3, a 2 = 0.5, a 3 = 0.8, k 1 = 0.001, k 2 = 0.0015, k 3 = 0.002 and b = 1.35 and . The initial conditions are given by x 0 = 0, y 0 = 0, and z 0 = 1.

Figure 2. A comparison between continuous and seasonal breeding with period two birth rate for model (1). The survivorship functions are s 1(x) = a 1/(1 + k 1 x), s 2(y) = a 2/(1 + k 2 y), s 3(z) = a 3/(1 + k 3 z) with parameter values a 1 = 0.3, a 2 = 0.5, a 3 = 0.8, k 1 = 0.001, k 2 = 0.0015, k 3 = 0.002 and b = 2 and . The initial conditions are given by x 0 = 0, y 0 = 0, and z 0 = 1.

3. Concluding remarks

In this paper we have shown that for the three-stage discrete-time population model (1) a periodic birth rate with period two is deleterious for low birth rates, as it will result in smaller average of adults in comparison with a continuous birth rate even though in both cases each adult reproduces the same number of juveniles per year. Numerical simulations of model (1) suggest that this conclusion remains true for large birth rates. This is in contrast with the result for a two-stage discrete model, which shows that when birth rates are low, seasonal reproduction is advantageous, while when birth rates are high continuous breeding is advantageous. The reason for this is that for a juvenile to become a breeding adult it has to survive two time units (two stages). Thus, a period two birth rate is not enough to compensate for this. Therefore, the seasonally breeding population needs to concentrate its breeding over a shorter time period. However, this period cannot be too short. If, for example, the seasonal population resorts to a period three birth rate, then numerical simulations of model (1) suggest that the conclusion is similar to that of the two-stage model with period two birth rate 13. That is, for low birth rates the three-stage model with period three birth rate results in higher adult averages than the continuous breeding population (see figure 3 for an example).

Figure 3. A comparison between continuous and seasonal breeding with period three birth rate for model (1). The survivorship functions are s 1(x) = a 1/(1 + k 1 x), s 2(y) = a 2/(1 + k 2 y), s 3(z) = a 3/(1 + k 3 z) with parameter values a 1 = 0.3, a 2 = 0.5, a 3 = 0.8, k 1 = 0.001, k 2 = 0.0015, k 3 = 0.002 and b = 1.5 and . The initial conditions are given by x 0 = 0, y 0 = 0 and z 0 = 1.

Similarly, if the birth rate has period four, then seasonal breeding results in higher adult averages for low birth rates. However, if the birth rate has period-five, then our numerical simulations suggest that seasonal breeding is always deleterious. Thus, the conclusion is similar to a period two birth rate.

If the population with three stages have a seasonal breeding of period three, i.e. b 0 = 0, b 1 = 0, , b 3 = 0, b 4 = 0 and then one can show that by defining the inherent net reproductive number for this population, a unique three-cycle exists provided that . We conjecture that this three-cycle is globally attracting provided that . While if then the population becomes extinct. These results should follow by using a similar argument as in section 2 (which involves lengthy computations). Thus, if (i.e. each individual reproduces in one season the same number of juveniles as an individual who reproduces continuously breeds in an entire year) then both populations persist if: However, since (recall that a 3 < 1) then for the seasonal population persists while the continuous breeding population goes to extinction. In figure 4 we present a numerical example for this case. Here, the inherent net reproductive numbers are R 0 = 0.9 and .

Figure 4. A comparison between continuous and seasonal breeding with period three birth rate for model (1). The survivorship functions are s 1(x) = a 1/(1 + k 1 x), s 2(y) = a 2/(1 + k 2 y), s 3(z) = a 3/(1 + k 3 z) with parameter values a 1 = 0.3, a 2 = 0.5, a 3 = 0.8, k 1 = 0.001, k 2 = 0.0015, k 3 = 0.002 and b = 1.2 and . The initial conditions are given by x 0 = 0, y 0 = 0 and z 0 = 1.

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Source: https://www.tandfonline.com/doi/full/10.1080/17513750701605440

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