Threshold dynamics of a predator–prey model with agestructured prey
 Yang Lu^{1} and
 Shengqiang Liu^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s136620181614y
© The Author(s) 2018
Received: 5 December 2017
Accepted: 24 April 2018
Published: 5 May 2018
Abstract
A predator–prey model, with aged structure in the prey population and the assumption that the predator hunts prey of all ages, is proposed and investigated. Using the uniform persistence theory for infinite dimensional dynamical systems, the global threshold dynamics of the model determined by the predator’s net reproductive number \(\Re_{P}\) are established: the predatorfree equilibrium is globally stable if \(\Re_{P}<1\), while the predator persists if \(\Re_{P}>1\). Numerical simulations are given to illustrate the results.
Keywords
MSC
1 Introduction
Predator–prey interactions are ubiquitous in the biological world, and they are one of the most important topics in ecology and continue to be of widespread interest today. Most existing studies on predator–prey models are focused on interacting species without age structure (see [1–4]). However, as the importance of age structure in populations has become more widely recognized, there is a rapidly growing literature dealing with various aspects of interacting populations with age structure [5–19].
In the above agestructured predator–prey population models, age structure is introduced into interactions of multispecies, and population models can quickly become remarkably complex [7, 13]. Hence, it is understandable that many studies in the dynamics of agestructured predator–prey populations assumed that age structure was only employed in one species, either in predators or prey [5, 13–16, 20]. When considering the age structure among the prey population, we can assume that predation is dependent on the age of the prey. This allows us to include agespecific predation into the model and to reflect on different possible settings from biology. In [13], a general framework for agestructured predator–prey systems is introduced. However, Mohr et al. [13] assumed that only adult prey is involved in predation. In [11], Li et al. argued that the prey population should have an age structure and they assumed that the functional response is of predatordependent type.
In this paper, we follow [8, 11, 13, 19, 21, 22] and propose a new predator–prey model with agestructured prey population. We assume that the predator hunts both the immature prey and the adult prey, and that the functional response of predators to prey is of Holling type II. Our primary aim of this paper is to obtain sharp criteria of the global threshold dynamics for the system.
The paper is organized as follows. In Sect. 2, we consider agestructured prey populations, define a threshold age, ageatmaturity, distinguish immature from adult individuals, and some assumption is introduced. In Sect. 3, we investigate the existence and stability of equilibria, and we find persistence. In the following section, we perform numerical simulations to verify our analytical results. At the end of the paper, we give a summary of the results.
2 The model

Let \(P(t)\) denote the total number of the predator at time t. Assume that the predator population is governed by the Lotka–Volterra equation. \(c(a)\) is the conversion efficiency of ingested prey into new predator individuals, and \(m(a)>0\) is the per capita capture rate of prey by a searching predator, \(h(a)>0\) is the handling (digestion) time per unit biomass consumed. In the absence of prey, the predator population, \(P(t)\), decreases exponentially with rate \(\mu_{P}>0\).

Let \(u(t,a)\) denote the prey population density of individuals of age a at time t. Biological interpretation suggests that \(\lim_{a\rightarrow+\infty}u(t,a)=0\), and we introduce a threshold age, \(\tau>0\), to distinguish immature individuals \((a<\tau)\) from adult ones \((a\geq\tau)\). Thus, we distinguish immature prey, \(u(t,a)=u_{1}{(t,a)}\), \(a\in[0,\tau)\), from adult prey, \(u(t,a)=u_{2}{(t,a)}\), \(a\in[\tau,+\infty)\). The transition from the immature class to the adult one occurs at age \(\tau>0\), the ageatmaturity of the prey. The total number of prey, \(U(t)\), is given by\(\mu:[0,+\infty)\rightarrow[0,+\infty)\) and \(\beta:[0,+\infty )\rightarrow[0,+\infty)\) denote the agedependent mortality and fertility rate of the prey, respectively. Here \(\beta(\cdot)\in L^{\infty}_{+}((0,+\infty),\mathbb{R})\) clearly describes the effects of the age on the fertility.$$U(t)= \int^{+\infty}_{0}u(t,a)\,da= \int^{\tau}_{0}u_{1}(t,a)\,da+ \int ^{+\infty}_{\tau} u_{2}(t,a) \,da=U_{1}(t)+U_{2}(t). $$
Variables and parameters used in the model
Symbol  Definition 

\(U_{1}(t)\)  number of juvenile prey at time t 
\(U_{2}(t)\)  number of mature prey at time t 
P(t)  number of the predator at time t 
\(u_{1}(t,a)\)  density of the juvenile prey at time t of age a 
\(u_{2}(t,a)\)  density of the mature prey at time t of age a 
\(\mu_{2}\)  per capita mortality rate of mature prey 
\(\mu_{1}\)  per capita mortality rate of juvenile prey 
\(\mu_{P}\)  per capita mortality rate for the predator 
\(c_{1}\)  prey juvenile biomass encounter rate 
\(c_{2}\)  prey adult biomass encounter rate 
β  the agespecific fertility rate or birth rate 
h(a)  the handling (digestion) time per unit biomass consumed 
\(m_{1}\)  the per capita capture rate of juvenile prey by a searching predator 
\(m_{2}\)  the per capita capture rate of mature prey by a searching predator 
τ  maturation time of juvenile prey 
u(t,0)  egg laying rate of mature prey 
θ  maximum per capita female egg release 
\(f(U_{2})\)  the effects of the predation on the fertility of mature prey: \((\theta U_{2})/(1+\theta U_{2})\) 
3 Mathematical analysis
3.1 Positivity and boundedness
Throughout this section, we always assume that (2.2) holds.
Proposition 3.1
Suppose that (2.2) holds, then all the solutions of system (2.8) are nonnegative and bounded for all \(t\geq0\) on their respective initial intervals (3.1).
Proof
We suppose that \(u^{0}_{1}(a)\geq0\), \(a\geq0\) is known, we take the solution of (2.7) as history function for (2.8) and we obtain nonnegative solutions if \(u^{0}_{1}(a)\) is not known. Following [13, p. 100], we can obtain positivity of solutions.
3.2 Existence of the boundary equilibria
3.3 Persistence and stability analysis
In this section, we study the global stability of the predatorfree equilibrium \(E_{1}\) of (2.8). Our principal result in this section can be stated as follows.
Theorem 3.1
Let \(\Re_{P}:= \frac{1}{\mu_{P}} \{\frac {c_{1}m_{1}{{U^{\ast}_{1}}}(t)}{1+h_{1}m_{1}U^{\ast}(t)}+\frac{c_{2}m_{2}{{U^{\ast}_{2}}}(t)}{1+h_{2}m_{2}U^{\ast}(t)} \}\), where \(U^{\ast}(t)={{U^{\ast}_{1}}(t)+{U^{\ast}_{2}}(t)}\). The predatorfree equilibrium \(E_{1}\) of (2.8) is globally asymptotically stable if \(\Re_{P}<1\).
Proof
Theorem 3.2
Proof
To complete the proof of Theorem 3.2, we now need to prove the following two claims.
Claim 1
Claim 2
4 Numerical simulations
5 Summary and discussion
In this paper, we study a predator–prey system with stage structured on the prey. The predator hunts both the immature prey and the adult prey. We have developed a rigorous analysis of the model by applying the comparison theory of differential equations and uniform persistence theory. Global dynamics of the model are obtained and threshold dynamics determined by the predator’s net reproductive number \(\Re_{P}\) are established: the predators go extinct if \(\Re_{P}<1\); and predators persist if \(\Re _{P}>1\). Theorem 3.1 shows that the predatorfree equilibrium \(E_{1}\) of (2.8) is globally asymptotically stable if \(\Re_{P}<1\). That the predator P is uniformly persistent is also obtained in Theorem 3.2.
First, we have constructed the predator’s net reproductive number \(\Re _{P}\), and by applying the comparison theory of differential equations, we get the predatorfree equilibrium \(E_{1}\) of (2.8) is globally asymptotically stable if \(\Re_{P}<1\) (see Theorem 3.1).
Second, by applying the uniform persistence theory, the predator P is uniformly persistent is also obtained in Theorem 3.2 (see Fig. 1).
Besides the above systematic theoretical results for model (2.8), we also perform careful numerical simulations to support the theoretical results. The prey have stage structure and the highlights of this paper are the effects by delay τ. It is shown that of the immature prey τ largely determines stability of the immature prey and the predator, in addition τ increases from 8 to 12/15.5 to 18.5, and the predator may lose its stability and becomes increasingly unstable by enlarging the amplitude of the oscillation interval (see Fig. 3). Biologically, this means that a shorter immature prey maturation period is helpful to stabilize the system.
In this paper, the stability of the predator–prey coexistence equilibrium remains unclear, which we leave as our future work.
Declarations
Acknowledgements
The authors would like to express their gratitude to the referees and the editor for their constructive, scientific thorough suggestions and comments on the manuscript. SL is supported by the NNSF of China (No. 11471089 and No. 11771374).
Authors’ contributions
YL and SL performed the mathematical analysis of the model, wrote and typeset the manuscript, and conducted the numerical simulations and discussions. SL formulated the model. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Xiao, Y., Chen, L.: Modeling and analysis of a predator–prey model with disease in the prey. Math. Biosci. 171, 59–82 (2001) View ArticleMATHMathSciNetGoogle Scholar
 Liu, S., Chen, L., Liu, Z.: Extionction and permanence in nonautonomous competitive system with stage structure. J. Math. Anal. Appl. 274, 667–684 (2002) View ArticleMATHMathSciNetGoogle Scholar
 Fan, M., Kuang, Y.: Dynamics of a nonautonomous predator–prey system with the Beddington–Deangelis functional response. J. Math. Anal. Appl. 295, 15–39 (2004) View ArticleMATHMathSciNetGoogle Scholar
 Rui, X., Chaplain, M.A.J., Davidson, F.A.: Permanence and periodicity of a delayed ratiodependent predator–prey model with stagestructure. J. Math. Anal. Appl. 303, 602–621 (2005) View ArticleMATHMathSciNetGoogle Scholar
 Lu, Y., Pawelek, P.A., Liu, S.: A stagestructured predator–prey model with predation over juvenile prey. Appl. Math. Comput. 297, 115–130 (2017) MathSciNetGoogle Scholar
 Liu, S., Beretta, E.: A stagestructured predator–prey model of Beddington–DeAngelis type. SIAM J. Appl. Math. 66, 1101–1129 (2006) View ArticleMATHMathSciNetGoogle Scholar
 Gourley, S., Lou, Y.: A mathematical model for the spatial spread and biocontrol of the astan longhorned beetle. SIAM J. Appl. Math. 74, 864–884 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Fang, J., Gourley, S., Lou, Y.: Stagestructured models of intra and interspecific competition within age classes. J. Differ. Equ. 260, 1918–1953 (2016) View ArticleMATHMathSciNetGoogle Scholar
 Browne, C., Pilyugin, S.: Global analysis of agestructured withinhost virus model. Discrete Contin. Dyn. Syst., Ser. B 18, 1999–2017 (2013) View ArticleMATHMathSciNetGoogle Scholar
 Van Den Driessche, P., Wang, L., Zou, X.: Modeling diseases with latency and relapse. Math. Biosci. Eng. 4, 205–219 (2007) View ArticleMATHMathSciNetGoogle Scholar
 Li, J.: Dynamics of agestructured predator–prey population models. J. Math. Anal. Appl. 152, 399–415 (1990) View ArticleMATHMathSciNetGoogle Scholar
 Delgado, M., Becerra, M., Suarez, A.: Analysis of an agestructured predator–prey model with disease in the prey. Nonlinear Anal., Real World Appl. 7, 853–871 (2006) View ArticleMATHMathSciNetGoogle Scholar
 Mohr, M., Barbarossa, M.V., Kuttler, C.: Predator–prey interactions, age structures and delay equations. Math. Model. Nat. Phenom. 9, 92–107 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Liu, Z., Magal, P., Ruan, S.: Predator–prey interactions, age structures and delay equations. Discrete Contin. Dyn. Syst., Ser. B 21, 537–555 (2016) View ArticleMATHMathSciNetGoogle Scholar
 Tang, H., Liu, Z.: Hopf bifurcation for a predator–prey model with age structure. Appl. Math. Model. 40, 726–737 (2016) View ArticleMathSciNetGoogle Scholar
 Liu, Z., Li, N.: Stability and bifurcation in a predator–prey model with age structure and delays. J. Nonlinear Sci. 25, 937–957 (2015) View ArticleMATHMathSciNetGoogle Scholar
 Bocharov, G., Hadeler, K.P.: Structured population models, conservation laws, and delay equations. J. Differ. Equ. 168, 212–237 (2000) View ArticleMATHMathSciNetGoogle Scholar
 Fister, K., Lenhart, S.: Optimal harvesting in an agestructured predator–prey model. Appl. Math. Optim. 54, 1–15 (2006) View ArticleMATHMathSciNetGoogle Scholar
 Liu, S., Xie, X., Tang, J.: Competing population model with nonlinear intraspecific regulation and maturation delays. Int. J. Biomath. 5, 12600071–126000722 (2012) MATHMathSciNetGoogle Scholar
 Cushing, J., Saleem, M.: A predator–prey model with agestructure. J. Math. Biol. 14, 231–250 (1982) View ArticleMATHMathSciNetGoogle Scholar
 Li, Y., Wang, J., Sun, B., Tang, J., Xie, X., Pang, S.: Modeling and analysis of the secondary routine dose against measles in China. Adv. Differ. Equ. 2017, 89 (2017) View ArticleMathSciNetGoogle Scholar
 Qiu, L., Yao, F., Zhong, X.: Stability analysis of networked control systems with random time delays and packet dropouts modeled by Markov chains. J. Appl. Math. 2013, 715072 (2013) MathSciNetGoogle Scholar
 Sharpe, F.R., Lotka, A.J.: A problem in age distribution. Philos. Mag. Ser. 6 21, 435–438 (1911) View ArticleMATHGoogle Scholar
 Evans, L.C.: Partial Differential Equations. AMS, Providence (1998) MATHGoogle Scholar
 Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) View ArticleMATHGoogle Scholar
 Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Amer. Math. Soc., Providence (1995) MATHGoogle Scholar
 Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993) MATHGoogle Scholar
 Liu, S., Chen, L., Luo, G., Jiang, L.: Asymptotic behavior of competitive Lotka–Volterra system with stage structure. J. Math. Anal. Appl. 271, 124–138 (2002) View ArticleMATHMathSciNetGoogle Scholar
 Zhao, X.Q.: Dynamical System in Population Biology. Springer, New York (2003) View ArticleMATHGoogle Scholar