# American Institute of Mathematical Sciences

2014, 11(4): 823-839. doi: 10.3934/mbe.2014.11.823

## Coexistence and asymptotic stability in stage-structured predator-prey models

 1 Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403 2 Mathematics and Statistics Department, University of North Carolina Wilmington, Wilmington, NC 28403-5970, United States, United States

Received  January 2013 Revised  August 2013 Published  March 2014

In this paper we analyze the effects of a stage-structured predator-prey system where the prey has two stages, juvenile and adult. Three different models (where the juvenile or adult prey populations are vulnerable) are studied to evaluate the impacts of this structure to the stability of the system and coexistence of the species. We assess how various ecological parameters, including predator mortality rate and handling times on prey, prey growth rate and death rate, prey capture rate and nutritional values in two stages, affect the existence and stability of all possible equilibria in each of the models, as well as the ultimate bounds and the dynamics of the populations. The main focus of this paper is to find general conditions to ensure the presence and stability of the coexistence equilibrium where both the predator and prey can co-exist Through specific examples, we demonstrate the stability of the trivial and co-existence equilibrium as well as the dynamics in each system.
Citation: Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823
##### References:
 [1] P. A. Abrams and C. Quince, The impact of mortality rate on predator population size and stability in systems with stage-structured prey, Theoretical Population Biology, 68 (2005), 253-266. [2] B. Buonomo, D. Lacitignola and S. Rionero, Effect of prey growth and predator cannibalism rate on the stability of a structured population model, Nonlinear Analysis: Real World Applications, 11 (2010), 1170-1181. doi: 10.1016/j.nonrwa.2009.01.053. [3] L. Cai and X. Song, Permanence and stability of a predator-prey system with stage structure for predator, Journal of Computational and Applied Mathematics, 201 (2007), 356-366. doi: 10.1016/j.cam.2005.12.035. [4] R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh, 123 (1993), 533-559. doi: 10.1017/S0308210500025877. [5] F. Chen, Permanence of periodic Holling type predator-prey system with stage structure for prey, Applied Mathematics and Computation, 182 (2006), 1849-1860. doi: 10.1016/j.amc.2006.06.024. [6] D. Cooke and J. A. Leon, Stability of population growth determined by 2x2 Leslie Matrix with density dependent elements, Biometrics, 32 (1976), 435-442. doi: 10.2307/2529512. [7] J. M. Cushing and M. Saleem, A predator prey model with age structure, Journal of Mathematical Biology, 14 (1982), 231-250. doi: 10.1007/BF01832847. [8] M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39. doi: 10.1016/j.jmaa.2004.02.038. [9] W. Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609. doi: 10.1006/jmaa.1993.1371. [10] W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209. doi: 10.1080/00036819408840277. [11] W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects, J. Math. Anal. Appl., 206 (1997), 547-566. doi: 10.1006/jmaa.1997.5265. [12] W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model, Diff. Integ. Eq.s, 8 (1995), 617-626. [13] W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478. doi: 10.3934/cpaa.2011.10.1463. [14] A. Hastings, Age-dependent predation is not a simple process, I: Continuous time models, Theorertical Population Biology, 23 (1983), 347-362. doi: 10.1016/0040-5809(83)90023-0. [15] A. Leung, Limiting behaviour for a prey-predator model with diffusion and crowding effects, J. Math. Biol., 6 (1978), 87-93. doi: 10.1007/BF02478520. [16] B. C. Longstaff, The dynamics of collembolan population growth, Canadian Journal of Zoology, 55 (1977), 314-324. doi: 10.1139/z77-043. [17] C. Lu, W. Feng and X. Lu, Long-term survival in a 3-species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213. [18] G. A. Oster, Internal variables in population dynamics, In Levin, S.A. (Ed.), Lectures on Mathematics in the Life Sciences, 8 (1976), American Mathematical Society, Providence, RI, 37-68. [19] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, N.Y., 1992. [20] R. H. Smith and R. Mead, Age structure and stability in models of prey-predator systems, Theoretical Population Biology, 6 (1974), 308-322. doi: 10.1016/0040-5809(74)90014-8. [21] C. Sun, Y. Lin and M. Han, Dynamic analysis for a stage-structure competitive system with mature population harvesting, Chaos, Solitons & Fractals, 31 (2007), 380-390. doi: 10.1016/j.chaos.2005.09.057. [22] H. Wang, J. D. Nagy, O. Gilg and Y. Kuang, The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, 221 (2009), 1-10. doi: 10.1016/j.mbs.2009.06.004.

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##### References:
 [1] P. A. Abrams and C. Quince, The impact of mortality rate on predator population size and stability in systems with stage-structured prey, Theoretical Population Biology, 68 (2005), 253-266. [2] B. Buonomo, D. Lacitignola and S. Rionero, Effect of prey growth and predator cannibalism rate on the stability of a structured population model, Nonlinear Analysis: Real World Applications, 11 (2010), 1170-1181. doi: 10.1016/j.nonrwa.2009.01.053. [3] L. Cai and X. Song, Permanence and stability of a predator-prey system with stage structure for predator, Journal of Computational and Applied Mathematics, 201 (2007), 356-366. doi: 10.1016/j.cam.2005.12.035. [4] R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh, 123 (1993), 533-559. doi: 10.1017/S0308210500025877. [5] F. Chen, Permanence of periodic Holling type predator-prey system with stage structure for prey, Applied Mathematics and Computation, 182 (2006), 1849-1860. doi: 10.1016/j.amc.2006.06.024. [6] D. Cooke and J. A. Leon, Stability of population growth determined by 2x2 Leslie Matrix with density dependent elements, Biometrics, 32 (1976), 435-442. doi: 10.2307/2529512. [7] J. M. Cushing and M. Saleem, A predator prey model with age structure, Journal of Mathematical Biology, 14 (1982), 231-250. doi: 10.1007/BF01832847. [8] M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39. doi: 10.1016/j.jmaa.2004.02.038. [9] W. Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609. doi: 10.1006/jmaa.1993.1371. [10] W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209. doi: 10.1080/00036819408840277. [11] W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects, J. Math. Anal. Appl., 206 (1997), 547-566. doi: 10.1006/jmaa.1997.5265. [12] W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model, Diff. Integ. Eq.s, 8 (1995), 617-626. [13] W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478. doi: 10.3934/cpaa.2011.10.1463. [14] A. Hastings, Age-dependent predation is not a simple process, I: Continuous time models, Theorertical Population Biology, 23 (1983), 347-362. doi: 10.1016/0040-5809(83)90023-0. [15] A. Leung, Limiting behaviour for a prey-predator model with diffusion and crowding effects, J. Math. Biol., 6 (1978), 87-93. doi: 10.1007/BF02478520. [16] B. C. Longstaff, The dynamics of collembolan population growth, Canadian Journal of Zoology, 55 (1977), 314-324. doi: 10.1139/z77-043. [17] C. Lu, W. Feng and X. Lu, Long-term survival in a 3-species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213. [18] G. A. Oster, Internal variables in population dynamics, In Levin, S.A. (Ed.), Lectures on Mathematics in the Life Sciences, 8 (1976), American Mathematical Society, Providence, RI, 37-68. [19] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, N.Y., 1992. [20] R. H. Smith and R. Mead, Age structure and stability in models of prey-predator systems, Theoretical Population Biology, 6 (1974), 308-322. doi: 10.1016/0040-5809(74)90014-8. [21] C. Sun, Y. Lin and M. Han, Dynamic analysis for a stage-structure competitive system with mature population harvesting, Chaos, Solitons & Fractals, 31 (2007), 380-390. doi: 10.1016/j.chaos.2005.09.057. [22] H. Wang, J. D. Nagy, O. Gilg and Y. Kuang, The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, 221 (2009), 1-10. doi: 10.1016/j.mbs.2009.06.004.
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