# American Institute of Mathematical Sciences

October  2013, 18(8): 2123-2142. doi: 10.3934/dcdsb.2013.18.2123

## Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates

 1 Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, United States

Received  November 2011 Revised  January 2013 Published  July 2013

The per-capita growth rate of a species is influenced by density-independent, positive and negative density-dependent factors. These factors can lead to nonlinearity with a consequence that species may process multiple nontrivial equilibria in its single state (e.g., Allee effects). This makes the study of permanence of discrete-time multi-species population models very challenging due to the complex boundary dynamics. In this paper, we explore the permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates for the first time. We find a simple sufficient condition for guaranteeing the permanence of the system by applying and extending the ecological concept of the relative nonlinearity to estimate systems' external Lyapunov exponents. Our method allows us to fully characterize the effects of nonlinearities in the per-capita growth functions and implies that the fluctuated populations may devastate the permanence of systems and lead to multiple attractors. These results are illustrated with specific two species competition and predator-prey models with generic nonlinear per-capita growth functions. Finally, we discuss the potential biological implications of our results.
Citation: Yun Kang. Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123
##### References:
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Gatto, Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations,, Theoretical Population Biology, 48 (1995), 126. doi: 10.1006/tpbi.1995.1024. Google Scholar [8] A. Fonda, Uniformly persistent semidynamical systems,, Proceedings of the American Mathematical Society, 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2. Google Scholar [9] L. R. Fox, Cannibalism in natural populations,, Annual Review of Ecology Systems, 6 (1975), 87. doi: 10.1146/annurev.es.06.110175.000511. Google Scholar [10] H. I. Freedman and J. W.-H. So, Persistence in discrete semidynamical systems,, SIAM Journal on Mathematical Analysis, 20 (1989), 930. doi: 10.1137/0520062. Google Scholar [11] B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations,, SIAM Journal on Mathematical Analysis, 34 (2003), 1007. doi: 10.1137/S0036141001392815. Google Scholar [12] V. Hutson, A theorem on average Liapunov functions,, Monatshefte für Mathematik, 98 (1984), 267. doi: 10.1007/BF01540776. Google Scholar [13] J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type,, Journal of Mathematical Biology, 25 (1987), 553. doi: 10.1007/BF00276199. Google Scholar [14] C. G. Jones, J. H. Lawton and M. Shachak, Positive and negative effects of organisms as physical ecosystem engineers,, Ecology, 78 (1997), 1946. Google Scholar [15] Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model,, Journal of Biological Dynamics, 2 (2008), 89. doi: 10.1080/17513750801956313. Google Scholar [16] Y. Kang and P. Chesson, Relative nonlinearity and permanence,, Theoretical Population Biology, 78 (2010), 26. doi: 10.1016/j.tpb.2010.04.002. Google Scholar [17] Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects,, Journal of Mathematical Biology, 62 (2011), 925. doi: 10.1007/s00285-010-0359-3. Google Scholar [18] Y. Kang and D. Armbruster, Dispersal effects on a discrete two-patch model for plant-insect interactions,, Journal of Theoretical Biology, 268 (2011), 84. doi: 10.1016/j.jtbi.2010.09.033. Google Scholar [19] Y. Kang and Y. Abdul-Aziz, Weak Allee effects and species coexistence,, Nonlinear Analysis: Real World Applications, 12 (2011), 3329. doi: 10.1016/j.nonrwa.2011.05.031. Google Scholar [20] Y. Kang, Pre-images of invariant sets of a discrete-time two species competition model,, Journal of Difference Equations and Applications, 18 (2012), 1709. doi: 10.1080/10236198.2011.591390. Google Scholar [21] Y. Kang and H. Smith, Global dynamics of a discrete-time two-species Lottery-Ricker competition model,, Journal of Biological Dynamics, 6 (2012), 358. doi: 10.1080/17513758.2011.586064. Google Scholar [22] Y. Kang and C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal,, Journal of Biological Dynamics, 6 (2012), 50. doi: 10.1080/17513758.2012.713125. Google Scholar [23] Y. Kang, Scramble competition can rescue endangered species subject to strong Allee effects,, Mathematical Biosciences, 241 (2013), 75. doi: 10.1016/j.mbs.2012.09.002. Google Scholar [24] N. Knowlton, Thresholds and multiple stable states in Coral reef community dynamics,, Integrative and Comparative Biology, 32 (1992), 674. doi: 10.1093/icb/32.6.674. Google Scholar [25] R. Kon, Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points,, Journal of Mathematical Biology, 48 (2004), 57. doi: 10.1007/s00285-003-0224-8. Google Scholar [26] R. Kon, Multiple attractors in host-parasitoid interactions: Coexistence and extinction,, Mathematical Biosciences, 201 (2006), 172. doi: 10.1016/j.mbs.2005.12.010. Google Scholar [27] M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly,, Oikos, 82 (1998), 384. doi: 10.2307/3546980. Google Scholar [28] D. Ludwig, D. D. Jones and C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest,, Journal of Animal Ecology, 47 (1978), 315. doi: 10.2307/3939. Google Scholar [29] R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton University Press, (1973). Google Scholar [30] R. M. Peterman, A simple mechanism that causes collapsing stability regions in exploited salmonid populations,, Journal of the Fisheries Research Board of Canada, 34 (1977), 1130. doi: 10.1139/f77-170. Google Scholar [31] D. A. Rand, H. B. Wilson and J. M. McGlade, Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics,, Philosophical Transactions of The Royal Society B: Biological Sciences, 343 (1994), 261. Google Scholar [32] P. L. Salceanu and H. Smith, Lyapunov exponents and uniform weak normally repelling invariant sets,, in, 389 (2009), 17. doi: 10.1007/978-3-642-02894-6_2. Google Scholar [33] P. L. Salceanu and H. Smith, Lyapunov exponents and persistence in some discrete dynamical systems,, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 187. doi: 10.3934/dcdsb.2009.12.187. Google Scholar [34] S. J. Schreiber, Criteria for $C^r$ robust permanence,, Journal of Differential Equations, 162 (2000), 400. doi: 10.1006/jdeq.1999.3719. Google Scholar [35] S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar [36] R. M. Sibly and J. Hone, Population growth rate and its determinants: An overview,, Philosophical Transactions of The Royal Society B: Biological Sciences, 357 (2002), 1153. Google Scholar [37] A. R. Solow, L. Stone and I. Rozdilsky, A critical smoothing test for multiple equilibria,, Ecology, 84 (2003), 1459. doi: 10.1890/02-3014. Google Scholar [38] P. D. Spencer and J. S. Collie, Patterns of population variability in marine fish stocks,, Fisheries Oceanography, 6 (1997), 188. doi: 10.1046/j.1365-2419.1997.00039.x. Google Scholar [39] P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation,, Trends in Ecology & Evolution, 14 (1999), 401. doi: 10.1016/S0169-5347(99)01684-5. Google Scholar [40] P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?,, Oikos, 87 (1999), 185. doi: 10.2307/3547011. Google Scholar [41] D. Tilman, "Resource Competition and Community Structure,", Monographs in Population Biology, 17 (1982). Google Scholar [42] P. Yodzis, "Introduction to Theoretical Ecology,", Harper & Row, (1989). Google Scholar [43] Xiao-Qiang Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16 (2003). Google Scholar

show all references

##### References:
 [1] R. Arditi and H. R. Akcakaya, Underestimation of mutual interference of predators,, Oecologia, 83 (1990), 358. Google Scholar [2] E. Beltrami, A mathematical model of the brown tide,, Estuaries, 12 (1989), 13. doi: 10.2307/1351445. Google Scholar [3] S. M. Carpenter, Alternate states of ecosystems: evidence and its implications,, in, (2001), 357. Google Scholar [4] P. Chesson, Multispecies competition in variable environments,, Theoretical Population Biology, 45 (1994), 227. doi: 10.1006/tpbi.1994.1013. Google Scholar [5] P. Chesson, Mechanisms of maintenance of species diversity,, Annual Review of Ecology Systems, 31 (2000), 343. doi: 10.1146/annurev.ecolsys.31.1.343. Google Scholar [6] F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation,", Oxford Scholarship Online, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001. Google Scholar [7] R. Ferriere and M. Gatto, Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations,, Theoretical Population Biology, 48 (1995), 126. doi: 10.1006/tpbi.1995.1024. Google Scholar [8] A. Fonda, Uniformly persistent semidynamical systems,, Proceedings of the American Mathematical Society, 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2. Google Scholar [9] L. R. Fox, Cannibalism in natural populations,, Annual Review of Ecology Systems, 6 (1975), 87. doi: 10.1146/annurev.es.06.110175.000511. Google Scholar [10] H. I. Freedman and J. W.-H. So, Persistence in discrete semidynamical systems,, SIAM Journal on Mathematical Analysis, 20 (1989), 930. doi: 10.1137/0520062. Google Scholar [11] B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations,, SIAM Journal on Mathematical Analysis, 34 (2003), 1007. doi: 10.1137/S0036141001392815. Google Scholar [12] V. Hutson, A theorem on average Liapunov functions,, Monatshefte für Mathematik, 98 (1984), 267. doi: 10.1007/BF01540776. Google Scholar [13] J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type,, Journal of Mathematical Biology, 25 (1987), 553. doi: 10.1007/BF00276199. Google Scholar [14] C. G. Jones, J. H. Lawton and M. Shachak, Positive and negative effects of organisms as physical ecosystem engineers,, Ecology, 78 (1997), 1946. Google Scholar [15] Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model,, Journal of Biological Dynamics, 2 (2008), 89. doi: 10.1080/17513750801956313. Google Scholar [16] Y. Kang and P. Chesson, Relative nonlinearity and permanence,, Theoretical Population Biology, 78 (2010), 26. doi: 10.1016/j.tpb.2010.04.002. Google Scholar [17] Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects,, Journal of Mathematical Biology, 62 (2011), 925. doi: 10.1007/s00285-010-0359-3. Google Scholar [18] Y. Kang and D. Armbruster, Dispersal effects on a discrete two-patch model for plant-insect interactions,, Journal of Theoretical Biology, 268 (2011), 84. doi: 10.1016/j.jtbi.2010.09.033. Google Scholar [19] Y. Kang and Y. Abdul-Aziz, Weak Allee effects and species coexistence,, Nonlinear Analysis: Real World Applications, 12 (2011), 3329. doi: 10.1016/j.nonrwa.2011.05.031. Google Scholar [20] Y. Kang, Pre-images of invariant sets of a discrete-time two species competition model,, Journal of Difference Equations and Applications, 18 (2012), 1709. doi: 10.1080/10236198.2011.591390. Google Scholar [21] Y. Kang and H. Smith, Global dynamics of a discrete-time two-species Lottery-Ricker competition model,, Journal of Biological Dynamics, 6 (2012), 358. doi: 10.1080/17513758.2011.586064. Google Scholar [22] Y. Kang and C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal,, Journal of Biological Dynamics, 6 (2012), 50. doi: 10.1080/17513758.2012.713125. Google Scholar [23] Y. Kang, Scramble competition can rescue endangered species subject to strong Allee effects,, Mathematical Biosciences, 241 (2013), 75. doi: 10.1016/j.mbs.2012.09.002. Google Scholar [24] N. Knowlton, Thresholds and multiple stable states in Coral reef community dynamics,, Integrative and Comparative Biology, 32 (1992), 674. doi: 10.1093/icb/32.6.674. Google Scholar [25] R. Kon, Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points,, Journal of Mathematical Biology, 48 (2004), 57. doi: 10.1007/s00285-003-0224-8. Google Scholar [26] R. Kon, Multiple attractors in host-parasitoid interactions: Coexistence and extinction,, Mathematical Biosciences, 201 (2006), 172. doi: 10.1016/j.mbs.2005.12.010. Google Scholar [27] M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly,, Oikos, 82 (1998), 384. doi: 10.2307/3546980. Google Scholar [28] D. Ludwig, D. D. Jones and C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest,, Journal of Animal Ecology, 47 (1978), 315. doi: 10.2307/3939. Google Scholar [29] R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton University Press, (1973). Google Scholar [30] R. M. Peterman, A simple mechanism that causes collapsing stability regions in exploited salmonid populations,, Journal of the Fisheries Research Board of Canada, 34 (1977), 1130. doi: 10.1139/f77-170. Google Scholar [31] D. A. Rand, H. B. Wilson and J. M. McGlade, Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics,, Philosophical Transactions of The Royal Society B: Biological Sciences, 343 (1994), 261. Google Scholar [32] P. L. Salceanu and H. Smith, Lyapunov exponents and uniform weak normally repelling invariant sets,, in, 389 (2009), 17. doi: 10.1007/978-3-642-02894-6_2. Google Scholar [33] P. L. Salceanu and H. Smith, Lyapunov exponents and persistence in some discrete dynamical systems,, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 187. doi: 10.3934/dcdsb.2009.12.187. Google Scholar [34] S. J. Schreiber, Criteria for $C^r$ robust permanence,, Journal of Differential Equations, 162 (2000), 400. doi: 10.1006/jdeq.1999.3719. Google Scholar [35] S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar [36] R. M. Sibly and J. Hone, Population growth rate and its determinants: An overview,, Philosophical Transactions of The Royal Society B: Biological Sciences, 357 (2002), 1153. Google Scholar [37] A. R. Solow, L. Stone and I. Rozdilsky, A critical smoothing test for multiple equilibria,, Ecology, 84 (2003), 1459. doi: 10.1890/02-3014. Google Scholar [38] P. D. Spencer and J. S. Collie, Patterns of population variability in marine fish stocks,, Fisheries Oceanography, 6 (1997), 188. doi: 10.1046/j.1365-2419.1997.00039.x. Google Scholar [39] P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation,, Trends in Ecology & Evolution, 14 (1999), 401. doi: 10.1016/S0169-5347(99)01684-5. Google Scholar [40] P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?,, Oikos, 87 (1999), 185. doi: 10.2307/3547011. Google Scholar [41] D. Tilman, "Resource Competition and Community Structure,", Monographs in Population Biology, 17 (1982). Google Scholar [42] P. Yodzis, "Introduction to Theoretical Ecology,", Harper & Row, (1989). Google Scholar [43] Xiao-Qiang Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16 (2003). Google Scholar
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