- Research
- Open Access
A Lycaon pictus impulsive state feedback control model with Allee effect and continuous time delay
- Yaning Li^{1},
- Huidong Cheng^{1}Email authorView ORCID ID profile and
- Yanhui Wang^{1, 2}
https://doi.org/10.1186/s13662-018-1820-7
© The Author(s) 2018
- Received: 9 May 2018
- Accepted: 27 September 2018
- Published: 11 October 2018
Abstract
Allee effect (i.e. sparse effect) is active when the population density is small. Our purpose is to study such an effect of this phenomenon on population dynamics. We investigate an impulsive state feedback control single-population model with Allee effect and continuous delay. We first qualitatively analyze the singularity of this model. Then we obtain sufficient conditions for the existence of an order-one periodic orbit by the geometric theory of impulsive differential equations for the survival of endangered populations and obtain the uniqueness of an order-one periodic orbit by the monotonicity of the subsequent function. Furthermore, we prove the orbital asymptotic stability of an order-one periodic orbit using the geometric properties of successor functions to confirm the robustness of this control. Finally, we verify the correctness of our theoretical results by using some numerical simulations. Our results show that the release of artificial captive African wild dog (Lycaon pictus) can effectively protect the African wild dog population with Allee effect.
Keywords
- Qualitatively analysis
- Order-one periodic orbit
- Impulsive differential equation
- Allee effect
MSC
- 34C25
- 34D23
- 92B05
- 34A37
1 Introduction
The African wild dog is one of the endangered carnivorous species in South Africa. They are mainly distributed in parts of eastern and southern Africa. In the past few decades the habitat of African wild dog has been drastically reduced, and the population quantity has declined significantly [1]. Research suggests that there are many reasons for the sharp decline in the distribution and number of African wild dogs, which may include conflicts with humans, habitat loss, persecution and competition with other predators, genetic diversity of infectious diseases, and inbreeding depression [2]. The scholars analyzed the demographic data of endangered Lycaon pictus in Hluhluwe–Imfolozi Park in South Africa from 1980 to 2004. They found that the African wild dog population has an obvious Allee effect [3]. In 1931, W.C. Allee paid attention to the possibility of a positive relationship between individual aspects of fitness and population density [4]. In other words, when the species has a small population density, the death rate increases, and the birth rate decreases, then the risk of species extinction increases. There are many reasons for this phenomenon. For example, for an individual, it is difficult to seek spouses and difficult to resist enemies and inbreeding depression [5]. Therefore many researches focus on the Allee effect (sparse effect) and have done a lot of work in this direction [6–9].
The Allee effect has been observed in many species, such as plants, marine invertebrates, and mammals [8]. For these species, there exists a minimum survival threshold, which implies that it is not necessary to adopt any measure to intervene when the population density is above this threshold [10]. But once the population density is lower than the survival threshold, some corresponding control measures should be performed according to the state of the target species. So the threshold strategy is also called the state feedback control strategy. This strategy can be precisely described by an impulsive differential equation in mathematics [11].
In recent years, impulsive differential equations have been used in various fields [12–20], such as disease control and pharmacology [21–29], integrated pest management [30–38], microbial culture [38–43], and protection of endangered animals and plants [44–51]. For example, Zhang et al. [10] focused on a predator–prey model with impulsive state feedback control and assuming that the spraying pesticide and releasing the natural enemies are taken at different thresholds. Liang et al. [5] investigated a state-dependent impulsive control model for computer virus propagation under media coverage, and the results show that the media coverage can delay the spread of computer virus. Nie et al. [52] studied different types of chemostat ecosystems of microbial cultures. However, there are few studies on the use of impulsive control strategies to protect the endangered species.
In this work, all the parameters are positive, and \(\int _{0}^{\infty }e^{(-s)}\,ds=1\).
The paper is organized is as follows. In Sect. 2, we first analyze qualitatively the singularity of system (3) without impulse effect by the Bendixson–Dulac theory. Then we use the geometric theory of impulsive differential equations to prove the existence of a periodic orbit. The uniqueness of the periodic orbit of model (3) is proved by the monotonicity of subsequent functions. Meanwhile, the geometric properties of subsequent functions are used to prove the stability of the periodic orbit of system (3) in Sect. 3. In Sect. 4, we illustrate the correctness of the results obtained by some numerical simulations. Finally, we conclude our work.
2 Dynamic analysis of system (3)
2.1 Qualitative analysis of system (3)
Theorem 2.1
If \((H_{1})\) holds, then the point \(E^{*}(x^{*},y^{*})\) is locally asymptotically stable.
Proof
Thus the point \(E^{*}(x^{*},y^{*})\) is a locally asymptotically stable node or focus. □
Theorem 2.2
If \((H_{1})\) and \((H_{2})\) hold, then the point \(E^{*}(x^{*},y^{*})\) is globally asymptotically stable.
Proof
2.2 Dynamic analysis of an order-one periodic orbit of system (3)
2.2.1 System (3) has a unique periodic orbit
In this subsection, we study the existence and uniqueness of an order-one periodic orbit. We assume that the impulsive set \(M=\{(x,y)\in R_{2}^{+}|x=h,y\geq 0\}\) and phase set \(N=\{(x,y)\in R_{2}^{+}|x=h+p,y\geq 0\}\) are straight lines. The trajectory that starts from any point A is denoted by \(f(A,t)\). A positive equilibrium \(E^{*}(x^{*}, y^{*})\) is globally asymptotically stable if conditions \((H_{1})\) and \((H_{2})\) hold.
Theorem 2.3
Proof
Case 1. \(0< h< h+p\leq x^{*}\).
According to the biological background, we only have to study the dynamic behavior of such a system in the region \(\Phi =\{(x,y)|x\geq h,0< y\leq b_{M}-Kx\}\), where \(b_{M}\) is a sufficiently large constant satisfying \(\frac{d\chi }{dt} |_{\chi =0}<0\).
The trajectory \(f(N^{\prime},t)\) will have no intersection point with the impulsive set M. Thus, the trajectory starting from any point located on the segment \(\overline{FN^{\prime}}\subset N\) is attractive to the positive equilibrium point \(E^{*}\) by impulse effect. The orbit starting from any point that is above the point F is also attractive to the point \(E^{*}\) by several impulsive effects at most.
Case 2. \(0< h< x^{*}< h+p< K\)
If \(0< h< x^{*}< h+p< K\) holds, then the point \(E^{*}(x^{*},y^{*})\) is between the impulsive set M and the phase set N. The isoclinic line \(\dot{x}=0\) intersects the impulsive set M at the point \(F_{1}(h,y_{F_{1}})\). The impulsive set M intersects the x-axis at the point \(M^{\prime}(h,0)\), and the intersection point of the phase set N and the x-axis is the point \(N^{\prime}(h+p,0)\).
Similarly, the orbit that starts from any point on the segment \(\overline{FN^{\prime}}\subset N\) will tend to the positive equilibrium point \(E^{*}(x^{*},y^{*})\), and the trajectory starting from any point above the point F on the phase set N will also be attractive to the point \(E^{*}\) by several impulse effects at most.
Theorem 2.4
If \(0< x^{*}\leq h< h+p< K\), then system (3) has an order-one periodic orbit, and when \(h+p>\frac{\sqrt{\Lambda }+(brkca-\omega r-br)}{2(brca+\omega Kca)}\), the order-one periodic orbit is unique.
Proof
Now, we prove the uniqueness of the order-one periodic orbit of system (3).
Thus, the successor function is decreasing monotonously on the phase set N. So there must exist a unique point S such that \(g(S)=0\). This means that the order-one periodic orbit of system (3) is unique. □
3 Stability of the order-one periodic solution
Theorem 3.1
The periodic orbit of system (3) is orbitally asymptotically stable.
Proof
By Theorems 2.3 and 2.4 system (3) has a unique order-one periodic orbit between the points H and F, and \(y_{F}< y_{S}< y_{H}\). Thus \(g(F)>0\) for any \(F\in N\) with \(y_{S}>y_{F}\), \(g(H)<0\) for any \(H\in N\) with \(y_{H}>y_{S}\), and \(g(F)=g(H)=0\) if and only if \(H=F=S\).
Thus our theoretical results show that releasing the African wild dogs in captivity to the wild is effective for protecting African wild dog population with Allee effect and continuous delay. Therefore, when protecting African wild dogs, we can determine the survival threshold, have African wild dogs in captivity, and monitor the wild populations (the initial value). Then, according to the state feedback of wild white-headed langurs, a certain amount of white-headed langurs in captivity will be released to the wild to increase the number of white-headed langurs in the wild, making the population have a normal reproduction to survive. According to the survival threshold and initial value, we can choose different grazing plans.
4 Numerical simulations and conclusion
4.1 Numerical simulations
In this section, we verify the correctness of the results by two examples.
Example 4.1
Then we get the following cases.
Example 4.2
We verify the feasibility of feedback control strategy by a real-life example.
4.2 Conclusion
In this paper, we studied the African wild dogs impulsive state feedback control model with Allee effect and continuous time delay. By the feedback information of the density of African wild dog population from the monitor we can protect the African wild dog population.
First, we carried on the quantitative and qualitative analysis and obtained two conditions \((H_{1})\) and \((H_{2})\). We proved the global asymptotic stability of the positive equilibrium \(E^{*}(x^{*},y^{*})\) by the Bendixson–Dulac theory.
Then we proved that the existence of an order-one periodic orbit of system (3) by the geometric theory of differential equations. We also proved the uniqueness of the order-one periodic orbit of system (3) by the monotonicity of the successor functions and the Lagrange mean value theorem.
Finally, we studied the orbital asymptotic stability of the order-one periodic orbit by the geometric properties of successor functions. Then we proved that the limit exists by the uniqueness of the order-one periodic orbit and the limit existence theorem.
All the results suggest that the release of artificial captive of African wild dogs can effectively protect the African wild dog population with Allee effect. The determination of the value of the survival threshold h involves analyzing the viability of the African wild dog population, which will be our future work.
Declarations
Funding
The paper was supported by the National Natural Science Foundation of China (No. 11371230, 11501331), Shandong Provincial Natural Science Foundation, China (No. S2015SF002), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.
Authors’ contributions
All 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
- Lindsey, P.A., Alexander, R.R., Toit, J.T.D., Mills, M.G.L.: The potential contribution of ecotourism to African wild dog Lycaon pictus conservation in South Africa. Biol. Conserv. 123(3), 339–348 (2005) View ArticleGoogle Scholar
- Courchamp, F., Brock, T.C., Grenfell, B.: Multipack dynamics and the Allee effect in the African wild dog, lycaon pictus. Anim. Conserv. 3(4), 277–285 (2000) View ArticleGoogle Scholar
- Somers, M.J., Graf, J.A., Szykman, M., Slotow, R., Gusset, M.: Dynamics of a small re-introduced population of wild dogs over 25 years: Allee effects and the implications of sociality for endangered species’ recovery. Oecologia 158(2), 239 (2008) View ArticleGoogle Scholar
- Stephens, P.A., Sutherland, W.J., Freckleton, R.P.: What is the Allee effect? Oikos 87(1), 185–190 (1999) View ArticleGoogle Scholar
- Chen, S., Xu, W., Chen, L., Huang, Z.: A white-headed langurs impulsive state feedback control model with sparse effect and continuous delay. Commun. Nonlinear Sci. Numer. Simul. 50, 88–102 (2017) MathSciNetView ArticleGoogle Scholar
- Chen, L.: Pest control and geometric theory of semi-continuous dynamical system. J. Beihua Univ. Nat. Sci. 12(1), 1–12 (2011) MathSciNetGoogle Scholar
- Liu, Q., Huang, L., Chen, L.: A pest management model with state feedback control. Adv. Differ. Equ. 2016(1), 292 (2016) MathSciNetView ArticleGoogle Scholar
- Yu, X., Yuan, S., Zhang, T.: Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching. Commun. Nonlinear Sci. Numer. Simul. 59, 359–374 (2018) MathSciNetView ArticleGoogle Scholar
- Bian, F., Zhao, W., Song, Y., Yue, R.: Dynamical analysis of a class of prey–predator model with Beddington–DeAngelis functional response, stochastic perturbation, and impulsive toxicant input. Complexity 2017, Article ID 3742197 (2017). https://doi.org/10.1155/2017/3742197 MATHView ArticleGoogle Scholar
- Zhang, T., Ma, W., Meng, X., Zhang, T.: Periodic solution of a prey–predator model with nonlinear state feedback control. Appl. Math. Comput. 266, 95–107 (2015) MathSciNetGoogle Scholar
- Liang, Z., Zeng, X., Pang, G., Liang, Y.: Periodic solution of a Leslie predator-prey system with ratio-dependent and state impulsive feedback control. Nonlinear Dyn. 89(4), 2941–2955 (2017). https://doi.org/10.1007/s11071-017-3637-4 MathSciNetMATHView ArticleGoogle Scholar
- Li, Y., Cheng, H., Wang, J., Wang, Y.: Dynamic analysis of unilateral diffusion Gompertz model with impulsive control strategy. Adv. Differ. Equ. 2018(1), 32 (2018) MathSciNetView ArticleGoogle Scholar
- Wang, J., Cheng, H., Li, Y., Zhang, X.: The geometrical analysis of a predator–prey model with multi-state dependent impulsive. J. Appl. Anal. Comput. 8(2), 427–442 (2018) MathSciNetGoogle Scholar
- Cheng, H., Zhang, T., Wang, F.: Existence and attractiveness of order one periodic solution of a Holling I predator–prey model. Abstr. Appl. Anal. 2012, Article ID 126018 (2012). https://doi.org/10.1155/2012/126018 MathSciNetMATHView ArticleGoogle Scholar
- Liu, F.: Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequal. Appl. 21(1), 25–40 (2018) MathSciNetMATHGoogle Scholar
- Liu, B., Tian, Y., Kang, B.: Existence and attractiveness of order one periodic solution of a Holling II predator–prey model with state-dependent impulsive control. Int. J. Biomath. 5(03), 675 (2012) Google Scholar
- Huang, M., Song, X., Li, J.: Modelling and analysis of impulsive releases of sterile mosquitoes. J. Biol. Dyn. 11(1), 147 (2017) MathSciNetView ArticleGoogle Scholar
- Zhang, S., Meng, X., Wang, X.: Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function. Adv. Differ. Equ. 2018(1), 50 (2018) MathSciNetView ArticleGoogle Scholar
- Liu, F., Xue, Q., Yabuta, K.: Rough maximal singular integral and maximal operators supported by subvarieties on Triebel–Lizorkin spaces. Nonlinear Anal. 171, 41–72 (2018) MathSciNetMATHView ArticleGoogle Scholar
- Braverman, E., Liz, E.: Global stabilization of periodic orbits using a proportional feedback control with pulses. Nonlinear Dyn. 67(4), 2467–2475 (2012) MathSciNetMATHView ArticleGoogle Scholar
- Zhang, M., Song, G., Chen, L.: A state feedback impulse model for computer worm control. Nonlinear Dyn. 85(3), 1561–1569 (2016) MathSciNetMATHView ArticleGoogle Scholar
- Zhang, T., Meng, X., Song, Y., Zhang, T.: A stage-structured predator–prey SI model with disease in the prey and impulsive effects. Math. Model. Anal. 18(4), 505–528 (2013) MathSciNetMATHView ArticleGoogle Scholar
- Meng, X., Zhao, S., Feng, T., Zhang, T.: Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. J. Math. Anal. Appl. 433(1), 227–242 (2016) MathSciNetMATHView ArticleGoogle Scholar
- Miao, A., Wang, X., Zhang, T., Wang, W., Sampath Aruna Pradeep, B.: Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis. Adv. Differ. Equ. 2017(1), 226 (2017) MathSciNetView ArticleGoogle Scholar
- Guo, H., Chen, L., Song, X.: Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control. Int. J. Biomath. 10(7), 1750093 (2017). https://doi.org/10.1142/S1793524517500930 MathSciNetMATHView ArticleGoogle Scholar
- Wang, W., Zhang, T.: Caspase-1-mediated pyroptosis of the predominance for driving CD4^{+} T cells death: a nonlocal spatial mathematical model. Bull. Math. Biol. 80(3), 540–582 (2018) MathSciNetMATHView ArticleGoogle Scholar
- Leng, X., Feng, T., Meng, X.: Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps. J. Inequal. Appl. 2017(1), 138 (2017) MathSciNetMATHView ArticleGoogle Scholar
- Miao, A., Jian, Z., Zhang, T., Pradeep, B.G.S.A.: Threshold dynamics of a stochastic SIR model with vertical transmission and vaccination. Comput. Math. Methods Med. 2017, Article ID 4820183 (2017). https://doi.org/10.1155/2017/4820183 MathSciNetMATHView ArticleGoogle Scholar
- Li, F., Meng, X., Wang, X.: Analysis and numerical simulations of a stochastic SEIQR epidemic system with quarantine-adjusted incidence and imperfect vaccination. Comput. Math. Methods Med. 2018(2), 1–14 (2018) MathSciNetGoogle Scholar
- Cheng, H., Zhang, T.: A new predator–prey model with a profitless delay of digestion and impulsive perturbation on the prey. Appl. Math. Comput. 217(22), 9198–9208 (2011) MathSciNetMATHGoogle Scholar
- Wang, J., Cheng, H., Meng, X., Pradeep, B.S.A.: Geometrical analysis and control optimization of a predator–prey model with multi state-dependent impulse. Adv. Differ. Equ. 2017(1), 252 (2017) MathSciNetView ArticleGoogle Scholar
- Wang, J., Cheng, H., Liu, H., Wang, Y.: Periodic solution and control optimization of a prey–predator model with two types of harvesting. Adv. Differ. Equ. 2018(1), 41 (2018) MathSciNetView ArticleGoogle Scholar
- Liu, H., Cheng, H.: Dynamic analysis of a prey-predator model with state-dependent control strategy and square root response function. Adv. Differ. Equ. 2018(1), 63 (2018). https://doi.org/10.1186/s13662-018-1507-0 MathSciNetView ArticleGoogle Scholar
- Zhang, H., Chen, L., Georgescu, P.: Impulsive control strategies for pest management. J. Biol. Syst. 15(02), 235–260 (2007) MATHView ArticleGoogle Scholar
- Zhang, H., Jiao, J., Chen, L.: Pest management through continuous and impulsive control strategies. Biosystems 90(2), 350–361 (2007) View ArticleGoogle Scholar
- Jiang, G., Lu, Q.: Impulsive state feedback control of a predator-prey model. J. Comput. Appl. Math. 200(1), 193–207 (2007) MathSciNetMATHView ArticleGoogle Scholar
- Liu, X., Zhang, T., Meng, X., Zhang, T.: Turing-Hopf bifurcations in a predator-prey model with herd behavior, quadratic mortality and prey-taxis. Phys. A, Stat. Mech. Appl. 496, 446–460 (2018) MathSciNetView ArticleGoogle Scholar
- Lv, X., Wang, L., Meng, X.: Global analysis of a new nonlinear stochastic differential competition system with impulsive effect. Adv. Differ. Equ. 2017(1), 296 (2017) MathSciNetView ArticleGoogle Scholar
- Zhang, T., Ma, W., Meng, X.: Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input. Adv. Differ. Equ. 2017, 115 (2017) MathSciNetView ArticleGoogle Scholar
- Meng, X., Wang, L., Zhang, T.: Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment. J. Appl. Anal. Comput. 6(3), 865–875 (2016) MathSciNetGoogle Scholar
- Zhang, T., Zhang, T., Meng, X.: Stability analysis of a chemostat model with maintenance energy. Appl. Math. Lett. 68, 1–7 (2017) MathSciNetMATHView ArticleGoogle Scholar
- Chi, M., Zhao, W.: Dynamical analysis of multi-nutrient and single microorganism chemostat model in a polluted environment. Adv. Differ. Equ. 2018(1), 120 (2018) MathSciNetView ArticleGoogle Scholar
- Jiao, J., Cai, S., Liu, W., Li, L.: Dynamics of a competitive population system with impulsive reduction of the invasive population. Electron. J. Differ. Equ. 2017, Article ID 281 (2017) MathSciNetMATHView ArticleGoogle Scholar
- Zhang, T., Liu, X., Meng, X., Zhang, T.: Spatio-temporal dynamics near the steady state of a planktonic system. Comput. Math. Appl. 75(12), 4490–4504 (2018) MathSciNetView ArticleGoogle Scholar
- Pang, G., Chen, L.: Periodic solution of the system with impulsive state feedback control. Nonlinear Dyn. 78(1), 743–753 (2014) MathSciNetMATHView ArticleGoogle Scholar
- Lv, W., Wang, F., Li, Y.: Adaptive finite-time tracking control for nonlinear systems with unmodeled dynamics using neural networks. Adv. Differ. Equ. 2018(1), 159 (2018) MathSciNetView ArticleGoogle Scholar
- Tang, S., Chen, L.: Global attractivity in a food-limited population model with impulsive effects. J. Math. Anal. Appl. 292(1), 211–221 (2004) MathSciNetMATHView ArticleGoogle Scholar
- Zhuo, X.: Global attractability and permanence for a new stage-structured delay impulsive ecosystem. J. Appl. Anal. Comput. 8(2), 457–470 (2018) MathSciNetGoogle Scholar
- Tian, Y., Zhang, T., Sun, K.: Dynamics analysis of a pest management prey–predator model by means of interval state monitoring and control. Nonlinear Anal. Hybrid Syst. 23, 122–141 (2017) MathSciNetMATHView ArticleGoogle Scholar
- Ling, Z., Zhang, L., Zhu, M., Malay, B.: Dynamical behaviour of a generalist predator–prey model with free boundary. Bound. Value Probl. 2017(1), 139 (2017) MathSciNetView ArticleGoogle Scholar
- Zeng, G.Z., Chen, L.S., Chen, J.F.: Persistence and periodic orbits for two-species nonautonomous diffusion Lotka–Volterra models. Math. Comput. Model. 20(12), 69–80 (1994) MathSciNetMATHView ArticleGoogle Scholar
- Nie, L., Teng, Z., Hu, L.: The dynamics of a chemostat model with state dependent impulsive effects. Int. J. Bifurc. Chaos 21(05), 1311–1322 (2011) MathSciNetMATHView ArticleGoogle Scholar