Vol. 07  No. 09 ( 2018 ), Article ID: 26706 , 6 pages
10.12677/AAM.2018.79132

Note on the Global Asymptotic Stability of a Strengthening Type Predator-Prey Model with Stage Structure

Xiaoyan Huang, Hang Deng, Fengde Chen

College of Mathematics and Computer Science, Fuzhou University, Fuzhou Fujian

Received: Aug. 16th, 2018; accepted: Aug. 31st, 2018; published: Sep. 6th, 2018

ABSTRACT

A strengthening type predator-prey model with stage structure is revisited in this paper. We first show that the main results of the previous paper are incorrect. After that, by constructing some suitable Lyapunov functions, a set of sufficient conditions which ensure the globally asymptotically stable of the positive equilibrium is obtained. We show that the conditions which ensure the existence of the positive equilibrium are enough to ensure the globally asymptotically stable, and consequently, the system is permanent. Our results supplement and complement some known results.

Keywords:Stage Structure, Predator, Prey, Global Asymptotic Stability

1. 引言

$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=x\left(t\right)\left(r-bx\left(t\right)-a{y}_{2}\left(t\right)\right),\\ {\stackrel{˙}{y}}_{1}\left(t\right)=e{y}_{2}-\left({d}_{1}+c\right){y}_{1}\left(t\right),\\ {\stackrel{˙}{y}}_{2}\left(t\right)=e{y}_{1}\left(t\right)-{d}_{2}{y}_{2}\left(t\right)+kax\left(t\right){y}_{2}\left(t\right),\end{array}$ (1.1)

$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=x\left(t\right)\left(1-x\left(t\right)-{y}_{2}\left(t\right)\right),\\ {\stackrel{˙}{y}}_{1}\left(t\right)={y}_{2}\left(t\right)-{c}_{1}{y}_{1}\left(t\right),\\ {\stackrel{˙}{y}}_{2}\left(t\right)={c}_{2}{y}_{1}\left(t\right)-{c}_{3}{y}_{2}\left(t\right)+{c}_{4}x\left(t\right){y}_{2}\left(t\right),\end{array}$ (1.2)

$0<{c}_{3}-\frac{{c}_{2}}{{c}_{1}}<{c}_{4}$ (1.3)

${x}^{*}=\frac{{c}_{3}-\frac{{c}_{2}}{{c}_{1}}}{{c}_{4}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{1}^{*}=\frac{{y}_{2}^{*}}{{c}_{1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{2}^{*}=1-{x}^{*}.$ (1.4)

 (1.5)

${c}_{3}-\frac{{c}_{1}}{{c}_{2}}>{c}_{4}$ (1.6)

${c}_{3}>{c}_{4}+\frac{{c}_{1}}{{c}_{2}}+{c}_{4}{y}_{1}^{*}$ (1.7)

2. 主要结果

$V\left(t\right)={K}_{1}\left(x-{x}^{*}-{x}^{*}\mathrm{ln}\frac{x}{{x}^{*}}\right)+{K}_{2}\left({y}_{1}-{y}_{1}^{*}-{y}_{1}^{*}\mathrm{ln}\frac{{y}_{1}}{{y}_{1}^{*}}\right)+{K}_{3}\left({y}_{2}-{y}_{2}^{*}-{y}_{2}^{*}\mathrm{ln}\frac{{y}_{2}}{{y}_{2}^{*}}\right)$ (2.1)

$\begin{array}{l}1-{x}^{*}-{y}_{2}^{*}=0,\\ {y}_{2}^{*}-{c}_{1}{y}_{1}^{*}=0,\\ {c}_{2}{y}_{1}^{*}-{c}_{3}{y}_{2}^{*}+{c}_{4}{x}^{*}{y}_{2}^{*}=0.\end{array}$ (2.2)

$\begin{array}{c}{D}^{+}V\left(t\right)={K}_{1}\frac{x-{x}^{*}}{x}x\left(1-x-{y}_{2}\right)+{K}_{2}\frac{{y}_{1}-{y}_{1}^{*}}{{y}_{1}}\left({y}_{2}-{c}_{1}{y}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{3}\frac{{y}_{2}-{y}_{2}^{*}}{{y}_{2}}\left({c}_{2}{y}_{1}-{c}_{3}{y}_{2}+{c}_{4}x{y}_{2}\right)\\ ={K}_{1}\left(x-{x}^{*}\right)\left({x}^{*}+{y}_{2}^{*}-x-{y}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{2}\frac{{y}_{1}-{y}_{1}^{*}}{{y}_{1}}\left(\frac{1}{{y}_{1}^{*}}\left({y}_{2}{y}_{1}^{*}-{y}_{1}{y}_{2}^{*}\right)+\frac{{y}_{1}{y}_{2}^{*}}{{y}_{1}^{*}}-{c}_{1}{y}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{3}\frac{{y}_{2}-{y}_{2}^{*}}{{y}_{2}}\left(\frac{{c}_{2}}{{y}_{2}^{*}}\left({y}_{1}{y}_{2}^{*}-{y}_{1}^{*}{y}_{2}\right)+\frac{{c}_{2}{y}_{1}^{*}{y}_{2}}{{y}_{2}^{*}}-{c}_{3}{y}_{2}+{c}_{4}x{y}_{2}\right)\end{array}$

$\begin{array}{l}=-{K}_{1}{\left(x-{x}^{*}\right)}^{2}+{K}_{1}\left(x-{x}^{*}\right)\left({y}_{2}^{*}-{y}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{2}\frac{{y}_{1}-{y}_{1}^{*}}{{y}_{1}}\left(\frac{1}{{y}_{1}^{*}}\left({y}_{2}{y}_{1}^{*}-{y}_{2}{y}_{1}+{y}_{2}{y}_{1}-{y}_{1}{y}_{2}^{*}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{3}\frac{{y}_{2}-{y}_{2}^{*}}{{y}_{2}}\left(\frac{{c}_{2}}{{y}_{2}^{*}}\left({y}_{1}{y}_{2}^{*}-{y}_{1}{y}_{2}+{y}_{1}{y}_{2}-{y}_{1}^{*}{y}_{2}\right)-{c}_{4}{x}^{*}{y}_{2}+{c}_{4}x{y}_{2}\right)\end{array}$

$\begin{array}{l}=-{K}_{1}{\left(x-{x}^{*}\right)}^{2}+{K}_{1}\left(x-{x}^{*}\right)\left({y}_{2}^{*}-{y}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{2}\frac{{y}_{1}-{y}_{1}^{*}}{{y}_{1}}\left(\frac{1}{{y}_{1}^{*}}\left({y}_{2}{y}_{1}^{*}-{y}_{2}{y}_{1}+{y}_{2}{y}_{1}-{y}_{1}{y}_{2}^{*}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{3}\frac{{y}_{2}-{y}_{2}^{*}}{{y}_{2}}\left(\frac{{c}_{2}}{{y}_{2}^{*}}\left({y}_{1}{y}_{2}^{*}-{y}_{1}{y}_{2}+{y}_{1}{y}_{2}-{y}_{1}^{*}{y}_{2}\right)-{c}_{4}{x}^{*}{y}_{2}+{c}_{4}x{y}_{2}\right)\end{array}$

$\begin{array}{l}=-{K}_{1}{\left(x-{x}^{*}\right)}^{2}+{K}_{1}\left(x-{x}^{*}\right)\left({y}_{2}^{*}-{y}_{2}\right)+{K}_{2}\frac{{y}_{1}-{y}_{1}^{*}}{{y}_{1}}\frac{{y}_{2}}{{y}_{1}^{*}}\left({y}_{1}^{*}-{y}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{2}\frac{{y}_{1}-{y}_{1}^{*}}{{y}_{1}}\frac{{y}_{1}}{{y}_{1}^{*}}\left({y}_{2}-{y}_{2}^{*}\right)+{K}_{3}\frac{{y}_{2}-{y}_{2}^{*}}{{y}_{2}}\frac{{c}_{2}}{{y}_{2}^{*}}{y}_{1}\left({y}_{2}^{*}-{y}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{K}_{3}\frac{{y}_{2}-{y}_{2}^{*}}{{y}_{2}}\frac{{c}_{2}}{{y}_{2}^{*}}{y}_{2}\left({y}_{1}-{y}_{1}^{*}\right)+{K}_{3}\frac{{y}_{2}-{y}_{2}^{*}}{{y}_{2}}{y}_{2}\left(-{c}_{4}{x}^{*}+{c}_{4}x\right)\end{array}$

$\begin{array}{l}=-{K}_{1}{\left(x-{x}^{*}\right)}^{2}-{K}_{1}\left(x-{x}^{*}\right)\left({y}_{2}-{y}_{2}^{*}\right)-\frac{{K}_{2}}{{y}_{1}^{*}}\frac{{y}_{2}}{{y}_{1}}{\left({y}_{1}-{y}_{1}^{*}\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{{K}_{2}}{{y}_{1}^{*}}\left({y}_{1}-{y}_{1}^{*}\right)\left({y}_{2}-{y}_{2}^{*}\right)-\frac{{K}_{3}{c}_{2}}{{y}_{2}^{*}}\frac{{y}_{1}}{{y}_{2}}{\left({y}_{2}-{y}_{2}^{*}\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{{K}_{3}{c}_{2}}{{y}_{2}^{*}}\left({y}_{2}-{y}_{2}^{*}\right)\left({y}_{1}-{y}_{1}^{*}\right)+{K}_{3}{c}_{4}\left({y}_{2}-{y}_{2}^{*}\right)\left(x-{x}^{*}\right)\\ =-{K}_{1}{\left(x-{x}^{*}\right)}^{2}+\left({K}_{3}{c}_{4}-{K}_{1}\right)\left(x-{x}^{*}\right)\left({y}_{2}-{y}_{2}^{*}\right)-\frac{{K}_{2}}{{y}_{1}^{*}}\frac{{y}_{2}}{{y}_{1}}{\left({y}_{1}-{y}_{1}^{*}\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left(\frac{{K}_{2}}{{y}_{1}^{*}}+\frac{{K}_{3}{c}_{2}}{{y}_{2}^{*}}\right)\left({y}_{1}-{y}_{1}^{*}\right)\left({y}_{2}-{y}_{2}^{*}\right)-\frac{{K}_{3}{c}_{2}}{{y}_{2}^{*}}\frac{{y}_{1}}{{y}_{2}}{\left({y}_{2}-{y}_{2}^{*}\right)}^{2}\end{array}$ (2.3)

${K}_{1}={c}_{4},{K}_{2}=\frac{{c}_{2}{y}_{1}^{*}}{{y}_{2}^{*}},{K}_{3}=1,$ (2.4)

 (2.5)

3. 结论

Note on the Global Asymptotic Stability of a Strengthening Type Predator-Prey Model with Stage Structure[J]. 应用数学进展, 2018, 07(09): 1141-1146. https://doi.org/10.12677/AAM.2018.79132

1. 1. Chen, F.D., Chen, W.L., et al. (2013) Permanece of a Stage-Structured Predator-Prey System. Applied Mathematics and Computation, 219, 8856-8862.

2. 2. Chen, F.D., Xie, X.D., et al. (2012) Partial Survival and Extinction of a Delayed Predator-Prey Model with Stage Structure. Applied Mathematics and Computation, 219, 4157-4162.

3. 3. Chen, F.D., Wang, H.N., et al. (2013) Global Stability of a Stage-Structured Predator-Prey System. Applied Mathematics and Computation, 22, 345-353.

4. 4. Li, T.T., Chen, F.D., et al. (2017) Stability of a Mutualism Model in Plant-Pollinator System with Stage-Structure and the Beddington-De Angelis Functional Response. Journal of Nonlinear Functional Analysis, 2017, Article ID: 50.

5. 5. Li, Z., Han, M.A., et al. (2012) Global Stability of Stage-Structured Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes. International Journal of Biomathematics, 6, Article ID: 1250057.

6. 6. Li, Z., Han, M., et al. (2014) Global Stability of a Predator-Prey System with Stage Structure and Mutual Interference. Discrete and Continuous Dynamical Systems-Series B (DCDS-B), 19, 173-187.

7. 7. Lin, X., Xie, X., et al. (2016) Convergences of a Stage-Structured Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes. Advances in Difference Equations, 2016, 181.

8. 8. Pu, L.Q., Miao, Z.S., et al. (2015) Global Stability of a Stage-Structured Predator-Prey Model. Communications in Mathematical Biology and Neuroscience, 2015, Article ID: 5.

9. 9. Han, R.Y., Yang, L.Y., et al. (2015) Global Attractivity of a Single Species Stage-Structured Model with Feedback Control and Infinite Delay. Communications in Mathematical Biology and Neuroscience, 2015, Article ID: 6.

10. 10. Wu, H.L. and Chen, F.D. (2009) Harvesting of a Single-Species System Incorporating Stage Structure and Toxicity. Discrete Dynamics in Nature and Society, 2009, Article ID: 290123.

11. 11. Khajanchi, S. and Banerjee, S. (2017) Role of Constant Prey Refuge on Stage Structure Predator-Prey Model with Ratio Dependent Functional Response. Applied Mathematics and Computation, 314, 193-198.

12. 12. Wei, F. and Fu, Q. (2016) Globally Asymptotic Stability of a Predator-Prey Model with Stage Structure Incorporating Prey Refuge. International Journal of Biomathematics, 9, Article ID: 1650058.

13. 13. Xue, Y., Pu, L., et al. (2015) Global Stability of a Predator-Prey System with Stage Structure of Distributed-Delay Type. Communications in Mathematical Biology and Neuroscience, 2015, Article ID: 12.

14. 14. Naji, R.K. and Majeed, S.J. (2016) The Dynamical Analysis of a Prey-Predator Model with a Refuge-Stage Structure Prey Population. International Journal of Differential Equations, 2016, Article ID: 2010464.

15. 15. Lu, Y., Pawelek, K.A., et al. (2017) A Stage-Structured Predator-Prey Model with Predation over Juvenile Prey. Applied Mathematics and Computation, 29, 7115-7130.

16. 16. Yu, Y.M., Zhang, Y. and Wang, W.D. (2001) The Asymptotic Behavior of a Strengthening Type Predator-Prey Model with Stage Structure. Journal of Southwest China Normal University (Natural Science), 26, 363-367. (In Chinese)

17. 17. Hao, P.M., Wang, X.C. and Wei, J.J. (2018) Hopf Bifurcation Analysis of a Diffusive Single Species Model with Stage Structure and Strong Allee Effect. Mathematics and Computers in Simulation, 153, 1-14.

18. 18. Li, Z. and Chen, F.D. (2009) Extinction in Periodic Competitive Stage-Structured Lotka-Volterra Model with the Effects of Toxic Substances. Journal of Computational and Applied Mathematics, 23, 1143-1153.