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 and Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123
References:
[1]

R. Arditi and H. R. Akcakaya, Underestimation of mutual interference of predators, Oecologia, 83 (1990), 358-361.

[2]

E. Beltrami, A mathematical model of the brown tide, Estuaries, 12 (1989), 13-17. doi: 10.2307/1351445.

[3]

S. M. Carpenter, Alternate states of ecosystems: evidence and its implications, in "Ecology: Achievement and Challenge" (eds. N. J. Huntley and S. A. Levin), M. C. Press, Blackwell, London, (2001), 357-383.

[4]

P. Chesson, Multispecies competition in variable environments, Theoretical Population Biology, 45 (1994), 227-276. doi: 10.1006/tpbi.1994.1013.

[5]

P. Chesson, Mechanisms of maintenance of species diversity, Annual Review of Ecology Systems, 31 (2000), 343-366. doi: 10.1146/annurev.ecolsys.31.1.343.

[6]

F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford Scholarship Online, Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[7]

R. Ferriere and M. Gatto, Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations, Theoretical Population Biology, 48 (1995), 126-171. doi: 10.1006/tpbi.1995.1024.

[8]

A. Fonda, Uniformly persistent semidynamical systems, Proceedings of the American Mathematical Society, 104 (1988), 111-116. doi: 10.1090/S0002-9939-1988-0958053-2.

[9]

L. R. Fox, Cannibalism in natural populations, Annual Review of Ecology Systems, 6 (1975), 87-106. doi: 10.1146/annurev.es.06.110175.000511.

[10]

H. I. Freedman and J. W.-H. So, Persistence in discrete semidynamical systems, SIAM Journal on Mathematical Analysis, 20 (1989), 930-938. doi: 10.1137/0520062.

[11]

B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM Journal on Mathematical Analysis, 34 (2003), 1007-1039. doi: 10.1137/S0036141001392815.

[12]

V. Hutson, A theorem on average Liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275. doi: 10.1007/BF01540776.

[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), 553-570. doi: 10.1007/BF00276199.

[14]

C. G. Jones, J. H. Lawton and M. Shachak, Positive and negative effects of organisms as physical ecosystem engineers, Ecology, 78 (1997), 1946-1957.

[15]

Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model, Journal of Biological Dynamics, 2 (2008), 89-101. doi: 10.1080/17513750801956313.

[16]

Y. Kang and P. Chesson, Relative nonlinearity and permanence, Theoretical Population Biology, 78 (2010), 26-35. doi: 10.1016/j.tpb.2010.04.002.

[17]

Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects, Journal of Mathematical Biology, 62 (2011), 925-973. doi: 10.1007/s00285-010-0359-3.

[18]

Y. Kang and D. Armbruster, Dispersal effects on a discrete two-patch model for plant-insect interactions, Journal of Theoretical Biology, 268 (2011), 84-97. doi: 10.1016/j.jtbi.2010.09.033.

[19]

Y. Kang and Y. Abdul-Aziz, Weak Allee effects and species coexistence, Nonlinear Analysis: Real World Applications, 12 (2011), 3329-3345. doi: 10.1016/j.nonrwa.2011.05.031.

[20]

Y. Kang, Pre-images of invariant sets of a discrete-time two species competition model, Journal of Difference Equations and Applications, 18 (2012), 1709-1733. doi: 10.1080/10236198.2011.591390.

[21]

Y. Kang and H. Smith, Global dynamics of a discrete-time two-species Lottery-Ricker competition model, Journal of Biological Dynamics, 6 (2012), 358-376. doi: 10.1080/17513758.2011.586064.

[22]

Y. Kang and C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal, Journal of Biological Dynamics, 6 (2012), 50-79. doi: 10.1080/17513758.2012.713125.

[23]

Y. Kang, Scramble competition can rescue endangered species subject to strong Allee effects, Mathematical Biosciences, 241 (2013), 75-87. doi: 10.1016/j.mbs.2012.09.002.

[24]

N. Knowlton, Thresholds and multiple stable states in Coral reef community dynamics, Integrative and Comparative Biology, 32 (1992), 674-682. doi: 10.1093/icb/32.6.674.

[25]

R. Kon, Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points, Journal of Mathematical Biology, 48 (2004), 57-81. doi: 10.1007/s00285-003-0224-8.

[26]

R. Kon, Multiple attractors in host-parasitoid interactions: Coexistence and extinction, Mathematical Biosciences, 201 (2006), 172-183. doi: 10.1016/j.mbs.2005.12.010.

[27]

M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly, Oikos, 82 (1998), 384-392. doi: 10.2307/3546980.

[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), 315-332. doi: 10.2307/3939.

[29]

R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1973.

[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), 1130-1142. doi: 10.1139/f77-170.

[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), 261-283.

[32]

P. L. Salceanu and H. Smith, Lyapunov exponents and uniform weak normally repelling invariant sets, in "Positive Systems," Lecture Notes in Control and Information Sciences, 389, Springer, Berlin, (2009), 17-27. doi: 10.1007/978-3-642-02894-6_2.

[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), 187-203. doi: 10.3934/dcdsb.2009.12.187.

[34]

S. J. Schreiber, Criteria for $C^r$ robust permanence, Journal of Differential Equations, 162 (2000), 400-426. doi: 10.1006/jdeq.1999.3719.

[35]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.

[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), 1153-1170.

[37]

A. R. Solow, L. Stone and I. Rozdilsky, A critical smoothing test for multiple equilibria, Ecology, 84 (2003), 1459-1463. doi: 10.1890/02-3014.

[38]

P. D. Spencer and J. S. Collie, Patterns of population variability in marine fish stocks, Fisheries Oceanography, 6 (1997), 188-204. doi: 10.1046/j.1365-2419.1997.00039.x.

[39]

P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401-405. doi: 10.1016/S0169-5347(99)01684-5.

[40]

P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190. doi: 10.2307/3547011.

[41]

D. Tilman, "Resource Competition and Community Structure," Monographs in Population Biology, 17, Princeton University Press, Princeton, NJ, 1982.

[42]

P. Yodzis, "Introduction to Theoretical Ecology," Harper & Row, Publishers, New York, 1989.

[43]

Xiao-Qiang Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.

show all references

References:
[1]

R. Arditi and H. R. Akcakaya, Underestimation of mutual interference of predators, Oecologia, 83 (1990), 358-361.

[2]

E. Beltrami, A mathematical model of the brown tide, Estuaries, 12 (1989), 13-17. doi: 10.2307/1351445.

[3]

S. M. Carpenter, Alternate states of ecosystems: evidence and its implications, in "Ecology: Achievement and Challenge" (eds. N. J. Huntley and S. A. Levin), M. C. Press, Blackwell, London, (2001), 357-383.

[4]

P. Chesson, Multispecies competition in variable environments, Theoretical Population Biology, 45 (1994), 227-276. doi: 10.1006/tpbi.1994.1013.

[5]

P. Chesson, Mechanisms of maintenance of species diversity, Annual Review of Ecology Systems, 31 (2000), 343-366. doi: 10.1146/annurev.ecolsys.31.1.343.

[6]

F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford Scholarship Online, Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[7]

R. Ferriere and M. Gatto, Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations, Theoretical Population Biology, 48 (1995), 126-171. doi: 10.1006/tpbi.1995.1024.

[8]

A. Fonda, Uniformly persistent semidynamical systems, Proceedings of the American Mathematical Society, 104 (1988), 111-116. doi: 10.1090/S0002-9939-1988-0958053-2.

[9]

L. R. Fox, Cannibalism in natural populations, Annual Review of Ecology Systems, 6 (1975), 87-106. doi: 10.1146/annurev.es.06.110175.000511.

[10]

H. I. Freedman and J. W.-H. So, Persistence in discrete semidynamical systems, SIAM Journal on Mathematical Analysis, 20 (1989), 930-938. doi: 10.1137/0520062.

[11]

B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM Journal on Mathematical Analysis, 34 (2003), 1007-1039. doi: 10.1137/S0036141001392815.

[12]

V. Hutson, A theorem on average Liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275. doi: 10.1007/BF01540776.

[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), 553-570. doi: 10.1007/BF00276199.

[14]

C. G. Jones, J. H. Lawton and M. Shachak, Positive and negative effects of organisms as physical ecosystem engineers, Ecology, 78 (1997), 1946-1957.

[15]

Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model, Journal of Biological Dynamics, 2 (2008), 89-101. doi: 10.1080/17513750801956313.

[16]

Y. Kang and P. Chesson, Relative nonlinearity and permanence, Theoretical Population Biology, 78 (2010), 26-35. doi: 10.1016/j.tpb.2010.04.002.

[17]

Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects, Journal of Mathematical Biology, 62 (2011), 925-973. doi: 10.1007/s00285-010-0359-3.

[18]

Y. Kang and D. Armbruster, Dispersal effects on a discrete two-patch model for plant-insect interactions, Journal of Theoretical Biology, 268 (2011), 84-97. doi: 10.1016/j.jtbi.2010.09.033.

[19]

Y. Kang and Y. Abdul-Aziz, Weak Allee effects and species coexistence, Nonlinear Analysis: Real World Applications, 12 (2011), 3329-3345. doi: 10.1016/j.nonrwa.2011.05.031.

[20]

Y. Kang, Pre-images of invariant sets of a discrete-time two species competition model, Journal of Difference Equations and Applications, 18 (2012), 1709-1733. doi: 10.1080/10236198.2011.591390.

[21]

Y. Kang and H. Smith, Global dynamics of a discrete-time two-species Lottery-Ricker competition model, Journal of Biological Dynamics, 6 (2012), 358-376. doi: 10.1080/17513758.2011.586064.

[22]

Y. Kang and C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal, Journal of Biological Dynamics, 6 (2012), 50-79. doi: 10.1080/17513758.2012.713125.

[23]

Y. Kang, Scramble competition can rescue endangered species subject to strong Allee effects, Mathematical Biosciences, 241 (2013), 75-87. doi: 10.1016/j.mbs.2012.09.002.

[24]

N. Knowlton, Thresholds and multiple stable states in Coral reef community dynamics, Integrative and Comparative Biology, 32 (1992), 674-682. doi: 10.1093/icb/32.6.674.

[25]

R. Kon, Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points, Journal of Mathematical Biology, 48 (2004), 57-81. doi: 10.1007/s00285-003-0224-8.

[26]

R. Kon, Multiple attractors in host-parasitoid interactions: Coexistence and extinction, Mathematical Biosciences, 201 (2006), 172-183. doi: 10.1016/j.mbs.2005.12.010.

[27]

M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly, Oikos, 82 (1998), 384-392. doi: 10.2307/3546980.

[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), 315-332. doi: 10.2307/3939.

[29]

R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1973.

[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), 1130-1142. doi: 10.1139/f77-170.

[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), 261-283.

[32]

P. L. Salceanu and H. Smith, Lyapunov exponents and uniform weak normally repelling invariant sets, in "Positive Systems," Lecture Notes in Control and Information Sciences, 389, Springer, Berlin, (2009), 17-27. doi: 10.1007/978-3-642-02894-6_2.

[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), 187-203. doi: 10.3934/dcdsb.2009.12.187.

[34]

S. J. Schreiber, Criteria for $C^r$ robust permanence, Journal of Differential Equations, 162 (2000), 400-426. doi: 10.1006/jdeq.1999.3719.

[35]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.

[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), 1153-1170.

[37]

A. R. Solow, L. Stone and I. Rozdilsky, A critical smoothing test for multiple equilibria, Ecology, 84 (2003), 1459-1463. doi: 10.1890/02-3014.

[38]

P. D. Spencer and J. S. Collie, Patterns of population variability in marine fish stocks, Fisheries Oceanography, 6 (1997), 188-204. doi: 10.1046/j.1365-2419.1997.00039.x.

[39]

P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401-405. doi: 10.1016/S0169-5347(99)01684-5.

[40]

P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190. doi: 10.2307/3547011.

[41]

D. Tilman, "Resource Competition and Community Structure," Monographs in Population Biology, 17, Princeton University Press, Princeton, NJ, 1982.

[42]

P. Yodzis, "Introduction to Theoretical Ecology," Harper & Row, Publishers, New York, 1989.

[43]

Xiao-Qiang Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.

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