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March  2020, 25(3): 1159-1167. doi: 10.3934/dcdsb.2019214

## Global stability of the predator-prey model with a sigmoid functional response

 1 Department of Mathematics, Alabama A & M University, Normal, AL 35762, USA 2 Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA

* Corresponding author: Yinshu Wu

Received  December 2018 Revised  May 2019 Published  September 2019

A predator-prey model with Sigmoid functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recent developed approach shows that, associated with the model, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing an analytic and geometrical analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium. With this crucial property we are able to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional response.

Citation: Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214
##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707–723. doi: 10.1002/bit.260100602.  Google Scholar [2] C. Castillo-Chavez, Z. Feng and W. Huang, Global dynamics of a plant-herbivore model with toxin-determined functional response, SIAM Appl. Math., 72 (2012), 1002-1020.  doi: 10.1137/110851614.  Google Scholar [3] S. H. Ding, On a kind of predator-prey system, SIAM J. Math. Anal., 20 (1989), 1426-1435.  doi: 10.1137/0520092.  Google Scholar [4] W. Ding and W. Huang, Global dynamics of a predator-prey model with general Holling type functional responses, Journal of Dynamics and Differential Equations, (2019), 1–14. doi: 10.1007/s10884-019-09755-0.  Google Scholar [5] J. W. Feng and X. W. Zen, The Global stability of predator-prey system of Gause-Type with Holling III functional response, Wuhan University J. Natural Sciences, 5 (2000), 271-277.  doi: 10.1007/BF02830133.  Google Scholar [6] A. Gasull and A. Guillamon, Non-existence of limit cycles for some predator-prey systems, Proceedings of Equadiff., World Scientific, Singapore, 91 (1993), 538–543.  Google Scholar [7] G. W. Harrison, Global stability of prodatorprey interactions, J. Math Biol, 8 (1979), 159-171.  doi: 10.1007/BF00279719.  Google Scholar [8] M. Hassel, J. Lawton and J. Beddington, Sigmoid functional responses by invertebrate predators and parasitoids, J. Animal Ecology, 46 (1977), 249-262.  doi: 10.2307/3959.  Google Scholar [9] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar [10] S. B. Hsu and T. W. Hwang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar [11] X. C. Huang, Uniqueness of limit cycles of generalized Liénard systems and predator-prey systems, J. Phys. A: Math. Gen., 21 (1988), L685–L691. doi: 10.1088/0305-4470/21/13/003.  Google Scholar [12] Z. Ma, S. Wang, T. Wang and H. Tang, Stability analysis of prey-predator system with Holling type functional response and prey refuge, Advances in Difference Equations, 2017 (2017), 12pp. doi: 10.1186/s13662-017-1301-4.  Google Scholar [13] L. Real, The kinetics of functional response, Am. Nat., 111 (1977), 289-300.  doi: 10.1086/283161.  Google Scholar [14] M. L. Rosenzweig and R. H. MacArthur, Graphic representation and stability conditions of predator-prey interactions, Amer. Nat., 97 (1963), 209-223.   Google Scholar [15] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896.  Google Scholar [16] E. Saez and E. Gonzalez-Olivares, Dynamics of predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.  Google Scholar [17] G. Seo and G. S. K. Wolkowicz, Existence of multiple limit cycles in a predator-prey model with arctan ($ax$) as functional response, Comm. Math. Anal., 18 (2015), 64-68.   Google Scholar [18] G. Seo and G. S. K. Wolkowicz, Sensitive of the dynamics of the general Rosenzweig-MacArthur model to the mathematical form of the functional response: a bifurcation theory approach, J. Math. Biol., 76 (2018), 1873-1906.  doi: 10.1007/s00285-017-1201-y.  Google Scholar [19] J. Sugie, R. Kohno and R. Miyazaki, On a predator-prey system of Holling Type, Proceedings of AMS, 125 (1997), 2041-2050.  doi: 10.1090/S0002-9939-97-03901-4.  Google Scholar [20] J. Sugie, K. Miyamoto and K. Morino, Absence of limit cycles of a predator-prey system with a sigmoid functional response, Appl. Math. Lett., 9 (1996), 85-90.  doi: 10.1016/0893-9659(96)00056-0.  Google Scholar [21] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM Appl. Math., 48 (1988), 592-606.  doi: 10.1137/0148033.  Google Scholar [22] H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of Predator-Prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682.  doi: 10.1137/S0036139901397285.  Google Scholar

show all references

##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707–723. doi: 10.1002/bit.260100602.  Google Scholar [2] C. Castillo-Chavez, Z. Feng and W. Huang, Global dynamics of a plant-herbivore model with toxin-determined functional response, SIAM Appl. Math., 72 (2012), 1002-1020.  doi: 10.1137/110851614.  Google Scholar [3] S. H. Ding, On a kind of predator-prey system, SIAM J. Math. Anal., 20 (1989), 1426-1435.  doi: 10.1137/0520092.  Google Scholar [4] W. Ding and W. Huang, Global dynamics of a predator-prey model with general Holling type functional responses, Journal of Dynamics and Differential Equations, (2019), 1–14. doi: 10.1007/s10884-019-09755-0.  Google Scholar [5] J. W. Feng and X. W. Zen, The Global stability of predator-prey system of Gause-Type with Holling III functional response, Wuhan University J. Natural Sciences, 5 (2000), 271-277.  doi: 10.1007/BF02830133.  Google Scholar [6] A. Gasull and A. Guillamon, Non-existence of limit cycles for some predator-prey systems, Proceedings of Equadiff., World Scientific, Singapore, 91 (1993), 538–543.  Google Scholar [7] G. W. Harrison, Global stability of prodatorprey interactions, J. Math Biol, 8 (1979), 159-171.  doi: 10.1007/BF00279719.  Google Scholar [8] M. Hassel, J. Lawton and J. Beddington, Sigmoid functional responses by invertebrate predators and parasitoids, J. Animal Ecology, 46 (1977), 249-262.  doi: 10.2307/3959.  Google Scholar [9] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar [10] S. B. Hsu and T. W. Hwang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar [11] X. C. Huang, Uniqueness of limit cycles of generalized Liénard systems and predator-prey systems, J. Phys. A: Math. Gen., 21 (1988), L685–L691. doi: 10.1088/0305-4470/21/13/003.  Google Scholar [12] Z. Ma, S. Wang, T. Wang and H. Tang, Stability analysis of prey-predator system with Holling type functional response and prey refuge, Advances in Difference Equations, 2017 (2017), 12pp. doi: 10.1186/s13662-017-1301-4.  Google Scholar [13] L. Real, The kinetics of functional response, Am. Nat., 111 (1977), 289-300.  doi: 10.1086/283161.  Google Scholar [14] M. L. Rosenzweig and R. H. MacArthur, Graphic representation and stability conditions of predator-prey interactions, Amer. Nat., 97 (1963), 209-223.   Google Scholar [15] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896.  Google Scholar [16] E. Saez and E. Gonzalez-Olivares, Dynamics of predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.  Google Scholar [17] G. Seo and G. S. K. Wolkowicz, Existence of multiple limit cycles in a predator-prey model with arctan ($ax$) as functional response, Comm. Math. Anal., 18 (2015), 64-68.   Google Scholar [18] G. Seo and G. S. K. Wolkowicz, Sensitive of the dynamics of the general Rosenzweig-MacArthur model to the mathematical form of the functional response: a bifurcation theory approach, J. Math. Biol., 76 (2018), 1873-1906.  doi: 10.1007/s00285-017-1201-y.  Google Scholar [19] J. Sugie, R. Kohno and R. Miyazaki, On a predator-prey system of Holling Type, Proceedings of AMS, 125 (1997), 2041-2050.  doi: 10.1090/S0002-9939-97-03901-4.  Google Scholar [20] J. Sugie, K. Miyamoto and K. Morino, Absence of limit cycles of a predator-prey system with a sigmoid functional response, Appl. Math. Lett., 9 (1996), 85-90.  doi: 10.1016/0893-9659(96)00056-0.  Google Scholar [21] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM Appl. Math., 48 (1988), 592-606.  doi: 10.1137/0148033.  Google Scholar [22] H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of Predator-Prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682.  doi: 10.1137/S0036139901397285.  Google Scholar
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