American Institute of Mathematical Sciences

Advanced
January  2021, 26(1): 633-644. doi: 10.3934/dcdsb.2020374

The $ P^* $ rule in the stochastic Holt-Lawton model of apparent competition

Sebastian J. Schreiber

Department of Evolution and Ecology, and Center for Population Biology, University of California, Davis, CA, 95616, USA

Received  July 2020 Revised  November 2020 Published  January 2021 Early access  December 2020

Fund Project: The author is supported by U.S. National Science Foundation grant DMS-1716803

In 1993, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species, stochastically varying parasitism rates, stochastically varying host intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the lack of direct, host density-dependence, we show that this system is dissipative i.e. enters a compact set in finite time for all initial conditions. When there is a single host species, stochastic persistence and extinction of the host is characterized using external Lyapunov exponents corresponding to the average per-capita growth rates of the host when rare. When a single host persists, say species $ i $, a explicit expression is derived for the average density, $ P_i^* $, of the parasitoid at the stationary distributions supporting both species. When there are multiple host species, we prove that the host species with the largest $ P_i^* $ value stochastically persists, while the other host species are asymptotically driven to extinction. A review of the main mathematical methods used to prove the results and future challenges are given.

Keywords: Apparent competition, environmental stochasticity, Lyapunov exponents, stochastic difference equations, exclusion.
Mathematics Subject Classification: Primary: 92D25, 60J05.
Citation: Sebastian J. Schreiber. The $ P^* $ rule in the stochastic Holt-Lawton model of apparent competition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 633-644. doi: 10.3934/dcdsb.2020374
References:
[1]

A. S. Ackleh, Y. M. Dib and S. R. J. Jang, Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model, Discrete and Continuous Dynamical Systems–Series B, 7 (2007), 683-698.  doi: 10.3934/dcdsb.2007.7.683.

[2]

A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality, Discrete and Continuous Dynamical Systems–Series B, 5 (2005), 175-188.  doi: 10.3934/dcdsb.2005.5.175.

[3]

M. Benaïm and S. J. Schreiber, Persistence and extinction for stochastic ecological models with internal and external variables, Journal of Mathematical Biology, 79 (2019), 393-431.  doi: 10.1007/s00285-019-01361-4.

[4]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.

[5]

M. B. Bonsall and M. P. Hassell, Apparent competition structures ecological assemblages, Nature, 388 (1997), 371-373.  doi: 10.1038/41084.

[6]

P. L. Chesson, The stabilizing effect of a random environment, Journal of Mathematical Biology, 15 (1982), 1-36.  doi: 10.1007/BF00275786.

[7]

F. Grognard, F. Mazenc and A. Rapaport, Polytopic Lyapunov functions for persistence analysis of competing species, Discrete and Continuous Dynamical Systems–Series B, 8 (2007), 73-93.  doi: 10.3934/dcdsb.2007.8.73.

[8]

M. P. Hassell, Host–parasitoid population dynamics, Journal of Animal Ecology, 69 (2000), 543-566.  doi: 10.1046/j.1365-2656.2000.00445.x.

[9]

M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology, vol. 13, Princeton University Press, Princeton, NJ, 1978.

[10]

M. P. Hassell, J. H. Lawton and J. R. Beddington, Sigmoid functional responses by invertebrate predators and parasitoids, The Journal of Animal Ecology, 46 (1977), 249-262.  doi: 10.2307/3959.

[11]

A. Hening, D. H. Nguyen and P. Chesson, A general theory of coexistence and extinction for stochastic ecological communities, preprint, 2020, arXiv: 2007.09025.

[12]

J. Hofbauer, A general cooperation theorem for hypercycles, Monatshefte für Mathematik, 91 (1981), 233-240.  doi: 10.1007/BF01301790.

[13] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.
[14]

R. D. Holt, Ecology at the mesoscale: The influence of regional processes on local communities, In Species Diversity in Ecological Communities, University of Chicago Press, Chicago, 1993, 77-88.

[15]

R. D. Holt, Predation, apparent competition and the structure of prey communities, Theoretical Population Biology, 12 (1977), 197-229.  doi: 10.1016/0040-5809(77)90042-9.

[16]

R. D. Holt and M. B. Bonsall, Apparent competition, Annual Review of Ecology, Evolution, and Systematics, 48 (2017), 447-471.  doi: 10.1146/annurev-ecolsys-110316-022628.

[17]

R. D. Holt and J. H. Lawton, Apparent competition and enemy-free space in insect host-parasitoid communities, American Naturalist, 142 (1993), 623-645.  doi: 10.1086/285561.

[18]

S. B. Hsu, M. C. Li, W. Liu and M. Malkin, Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss, Discrete and Continous Dynamical Systems, 9 (2003), 1465-1492.  doi: 10.3934/dcds.2003.9.1465.

[19]

S. B. Hsu and C. J. Lin, Dynamics of two phytoplankton species competing for light and nutrient with internal storage, Discrete and Continuous Dynamical Systems–Series S, 7 (2014), 1259-1285.  doi: 10.3934/dcdss.2014.7.1259.

[20]

S. B. Hsu and L. W. Roeger, A refuge-mediated apparent competition model, Canadian Applied Mathematical Quarterly, 19 (2011), 219-234. 

[21]

S. B. Hsu, J. Shi and and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.

[22]

T. B. Issa and R. B. Salako, Asymptotic dynamics in a two-species chemotaxis model with non-local terms, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 3839-3874.  doi: 10.3934/dcdsb.2017193.

[23]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.

[24] A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. 
[25]

Y. Kuang, J. D. Nagy and J. J. Elser, Biological stoichiometry of tumor dynamics: Mathematical models and analysis, Discrete and Continuous Dynamical Systems–Series B, 4 (2004), 221-240.  doi: 10.3934/dcdsb.2004.4.221.

[26]

M. R. S. Kulenović and O. Merino, Competitive-exclusion versus competitive-coexistence for systems in the plane, Discrete and Continuous Dynamical Systems–Series B., 6 (2006), 1141-1156.  doi: 10.3934/dcdsb.2006.6.1141.

[27]

C. J. Lin, Competition of two phytoplankton species for light with wavelength, Discrete and Continuous Dynamical Systems–Series B., 21 (2016), 523-536.  doi: 10.3934/dcdsb.2016.21.523.

[28]

J. Loman, A graphical solution to a one-predator, two-prey system with apparent competition and mutualism, Mathematical Biosciences, 91 (1988), 1-16.  doi: 10.1016/0025-5564(88)90021-1.

[29]

Y. Lou and D. Munther, Dynamics of a three species competition model, Discrete and Continuous Dynamical Systems–Series B, 32 (2012), 3099-3131.  doi: 10.3934/dcds.2012.32.3099.

[30]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.

[31]

R. M. May, Host-parasitoid systems in patch environments: A phenomenological model, Journal of Animal Ecology, 47 (1978), 833-843. 

[32]

R. M. May, M. P. Hassell, R. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, Journal of Animal Ecology, 50 (1981), 855-865.  doi: 10.2307/4142.

[33]

C. R. Miller, Y. Kuang, W. F. Fagan and J. J. Elser, Modeling and analysis of stoichiometric two-patch consumer-resource systems, Mathematical Biosciences, 189 (2004), 153-184.  doi: 10.1016/j.mbs.2004.01.004.

[34]

N. J. Mills and W. M. Getz, Modelling the biological control of insect pests: A review of host-parasitoid models, Ecological Modelling, 92 (1996), 121-143.  doi: 10.1016/0304-3800(95)00177-8.

[35]

R. J. Morris, O. T. Lewis and H. C. J. Godfray, Experimental evidence for apparent competition in a tropical forest food web, Nature, 428 (2004), 310-313.  doi: 10.1038/nature02394.

[36]

T. Namba, Dispersal-mediated coexistence of indirect competitors in source-sink metacommunities, Japan Journal of Industrial and Applied Mathematics, 24 (2007), 39-55.  doi: 10.1007/BF03167506.

[37]

A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proceedings of the Zoological Society of London, (1935), 551–598.

[38]

M. Núñez López, J. X. Velasco-Hernández and P. A. Marquet, The dynamics of technological change under constraints: Adopters and resources, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3299-3317.  doi: 10.3934/dcdsb.2014.19.3299.

[39]

H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3341-3357.  doi: 10.3934/dcdsb.2014.19.3341.

[40]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.

[41]

T. W. Schoener, The newest synthesis: Understanding the interplay of evolutionary and ecological dynamics, Science, 331 (2011), 426-429.  doi: 10.1126/science.1193954.

[42]

S. J. Schreiber, Coexistence for species sharing a predator, Journal of Differential Equations, 196 (2004), 209-225.  doi: 10.1016/S0022-0396(03)00169-4.

[43]

S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, Journal of Difference Equations and Applications, 18 (2012), 1381-1403.  doi: 10.1080/10236198.2011.628662.

[44]

S. J. Schreiber, When do factors promoting genetic diversity also promote population persistence? A demographic perspective on Gillespieś SAS-CFF model, Theoretical Population Biology, 133 (2020), 141-149. 

[45]

S. J. Schreiber, Coexistence in the Face of Uncertainty, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Springer, New York, 2017,349–384.

[46]

S. J. Schreiber and V. Křivan, Holt (1977) and apparent competition, Theoretical Population Biology, 133 (2020), 17-18.  doi: 10.1016/j.tpb.2019.09.006.

[47]

S. J. Schreiber, M. Benaïm and K. A. S. Atchadé, Persistence in fluctuating environments, Journal of Mathematical Biology, 62 (2011), 655-683.  doi: 10.1007/s00285-010-0349-5.

[48]

S. J. Schreiber, R. Bürger and D. I. Bolnick, The community effects of phenotypic and genetic variation within a predator population, Ecology, 92 (2011), 1582-1593.  doi: 10.1890/10-2071.1.

[49]

S. J. Schreiber and S. Patel, Evolutionarily induced alternative states and coexistence in systems with apparent competition, Natural Resource Modelling, 28 (2015), 475-496.  doi: 10.1111/nrm.12076.

[50]

H. L. Smith and X. Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete and Continuous Dynamical Systems–Series B, 1 (2001), 183-191.  doi: 10.3934/dcdsb.2001.1.183.

[51]

D. Tang, Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 4913-4928.  doi: 10.3934/dcdsb.2019037.

[52]

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

[53]

J. X. Velasco-Hernández, M. Núñez López, G. Ramírez-Santiago and M. Hernández-Rosales, On carrying-capacity construction, metapopulations and density-dependent mortality, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1099-1110.  doi: 10.3934/dcdsb.2017054.

[54]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.

[55]

H. Wang, K. Dunning, J. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete and Continuous Dynamical Systems–Series B, 12 (2009), 481-493.  doi: 10.3934/dcdsb.2009.12.481.

[56]

Y. Wu, N. Tuncer and M. Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1167-1187.  doi: 10.3934/dcdsb.2017057.

[57]

H. Yu, S. Zhong and R. P. Agarwal, Mathematics and dynamic analysis of an apparent competition community model with impulsive effect, Mathematical and Computer Modelling, 52 (2010), 25-36.  doi: 10.1016/j.mcm.2009.11.019.

show all references

References:
[1]

A. S. Ackleh, Y. M. Dib and S. R. J. Jang, Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model, Discrete and Continuous Dynamical Systems–Series B, 7 (2007), 683-698.  doi: 10.3934/dcdsb.2007.7.683.

[2]

A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality, Discrete and Continuous Dynamical Systems–Series B, 5 (2005), 175-188.  doi: 10.3934/dcdsb.2005.5.175.

[3]

M. Benaïm and S. J. Schreiber, Persistence and extinction for stochastic ecological models with internal and external variables, Journal of Mathematical Biology, 79 (2019), 393-431.  doi: 10.1007/s00285-019-01361-4.

[4]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.

[5]

M. B. Bonsall and M. P. Hassell, Apparent competition structures ecological assemblages, Nature, 388 (1997), 371-373.  doi: 10.1038/41084.

[6]

P. L. Chesson, The stabilizing effect of a random environment, Journal of Mathematical Biology, 15 (1982), 1-36.  doi: 10.1007/BF00275786.

[7]

F. Grognard, F. Mazenc and A. Rapaport, Polytopic Lyapunov functions for persistence analysis of competing species, Discrete and Continuous Dynamical Systems–Series B, 8 (2007), 73-93.  doi: 10.3934/dcdsb.2007.8.73.

[8]

M. P. Hassell, Host–parasitoid population dynamics, Journal of Animal Ecology, 69 (2000), 543-566.  doi: 10.1046/j.1365-2656.2000.00445.x.

[9]

M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology, vol. 13, Princeton University Press, Princeton, NJ, 1978.

[10]

M. P. Hassell, J. H. Lawton and J. R. Beddington, Sigmoid functional responses by invertebrate predators and parasitoids, The Journal of Animal Ecology, 46 (1977), 249-262.  doi: 10.2307/3959.

[11]

A. Hening, D. H. Nguyen and P. Chesson, A general theory of coexistence and extinction for stochastic ecological communities, preprint, 2020, arXiv: 2007.09025.

[12]

J. Hofbauer, A general cooperation theorem for hypercycles, Monatshefte für Mathematik, 91 (1981), 233-240.  doi: 10.1007/BF01301790.

[13] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.
[14]

R. D. Holt, Ecology at the mesoscale: The influence of regional processes on local communities, In Species Diversity in Ecological Communities, University of Chicago Press, Chicago, 1993, 77-88.

[15]

R. D. Holt, Predation, apparent competition and the structure of prey communities, Theoretical Population Biology, 12 (1977), 197-229.  doi: 10.1016/0040-5809(77)90042-9.

[16]

R. D. Holt and M. B. Bonsall, Apparent competition, Annual Review of Ecology, Evolution, and Systematics, 48 (2017), 447-471.  doi: 10.1146/annurev-ecolsys-110316-022628.

[17]

R. D. Holt and J. H. Lawton, Apparent competition and enemy-free space in insect host-parasitoid communities, American Naturalist, 142 (1993), 623-645.  doi: 10.1086/285561.

[18]

S. B. Hsu, M. C. Li, W. Liu and M. Malkin, Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss, Discrete and Continous Dynamical Systems, 9 (2003), 1465-1492.  doi: 10.3934/dcds.2003.9.1465.

[19]

S. B. Hsu and C. J. Lin, Dynamics of two phytoplankton species competing for light and nutrient with internal storage, Discrete and Continuous Dynamical Systems–Series S, 7 (2014), 1259-1285.  doi: 10.3934/dcdss.2014.7.1259.

[20]

S. B. Hsu and L. W. Roeger, A refuge-mediated apparent competition model, Canadian Applied Mathematical Quarterly, 19 (2011), 219-234. 

[21]

S. B. Hsu, J. Shi and and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.

[22]

T. B. Issa and R. B. Salako, Asymptotic dynamics in a two-species chemotaxis model with non-local terms, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 3839-3874.  doi: 10.3934/dcdsb.2017193.

[23]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.

[24] A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. 
[25]

Y. Kuang, J. D. Nagy and J. J. Elser, Biological stoichiometry of tumor dynamics: Mathematical models and analysis, Discrete and Continuous Dynamical Systems–Series B, 4 (2004), 221-240.  doi: 10.3934/dcdsb.2004.4.221.

[26]

M. R. S. Kulenović and O. Merino, Competitive-exclusion versus competitive-coexistence for systems in the plane, Discrete and Continuous Dynamical Systems–Series B., 6 (2006), 1141-1156.  doi: 10.3934/dcdsb.2006.6.1141.

[27]

C. J. Lin, Competition of two phytoplankton species for light with wavelength, Discrete and Continuous Dynamical Systems–Series B., 21 (2016), 523-536.  doi: 10.3934/dcdsb.2016.21.523.

[28]

J. Loman, A graphical solution to a one-predator, two-prey system with apparent competition and mutualism, Mathematical Biosciences, 91 (1988), 1-16.  doi: 10.1016/0025-5564(88)90021-1.

[29]

Y. Lou and D. Munther, Dynamics of a three species competition model, Discrete and Continuous Dynamical Systems–Series B, 32 (2012), 3099-3131.  doi: 10.3934/dcds.2012.32.3099.

[30]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.

[31]

R. M. May, Host-parasitoid systems in patch environments: A phenomenological model, Journal of Animal Ecology, 47 (1978), 833-843. 

[32]

R. M. May, M. P. Hassell, R. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, Journal of Animal Ecology, 50 (1981), 855-865.  doi: 10.2307/4142.

[33]

C. R. Miller, Y. Kuang, W. F. Fagan and J. J. Elser, Modeling and analysis of stoichiometric two-patch consumer-resource systems, Mathematical Biosciences, 189 (2004), 153-184.  doi: 10.1016/j.mbs.2004.01.004.

[34]

N. J. Mills and W. M. Getz, Modelling the biological control of insect pests: A review of host-parasitoid models, Ecological Modelling, 92 (1996), 121-143.  doi: 10.1016/0304-3800(95)00177-8.

[35]

R. J. Morris, O. T. Lewis and H. C. J. Godfray, Experimental evidence for apparent competition in a tropical forest food web, Nature, 428 (2004), 310-313.  doi: 10.1038/nature02394.

[36]

T. Namba, Dispersal-mediated coexistence of indirect competitors in source-sink metacommunities, Japan Journal of Industrial and Applied Mathematics, 24 (2007), 39-55.  doi: 10.1007/BF03167506.

[37]

A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proceedings of the Zoological Society of London, (1935), 551–598.

[38]

M. Núñez López, J. X. Velasco-Hernández and P. A. Marquet, The dynamics of technological change under constraints: Adopters and resources, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3299-3317.  doi: 10.3934/dcdsb.2014.19.3299.

[39]

H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3341-3357.  doi: 10.3934/dcdsb.2014.19.3341.

[40]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.

[41]

T. W. Schoener, The newest synthesis: Understanding the interplay of evolutionary and ecological dynamics, Science, 331 (2011), 426-429.  doi: 10.1126/science.1193954.

[42]

S. J. Schreiber, Coexistence for species sharing a predator, Journal of Differential Equations, 196 (2004), 209-225.  doi: 10.1016/S0022-0396(03)00169-4.

[43]

S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, Journal of Difference Equations and Applications, 18 (2012), 1381-1403.  doi: 10.1080/10236198.2011.628662.

[44]

S. J. Schreiber, When do factors promoting genetic diversity also promote population persistence? A demographic perspective on Gillespieś SAS-CFF model, Theoretical Population Biology, 133 (2020), 141-149. 

[45]

S. J. Schreiber, Coexistence in the Face of Uncertainty, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Springer, New York, 2017,349–384.

[46]

S. J. Schreiber and V. Křivan, Holt (1977) and apparent competition, Theoretical Population Biology, 133 (2020), 17-18.  doi: 10.1016/j.tpb.2019.09.006.

[47]

S. J. Schreiber, M. Benaïm and K. A. S. Atchadé, Persistence in fluctuating environments, Journal of Mathematical Biology, 62 (2011), 655-683.  doi: 10.1007/s00285-010-0349-5.

[48]

S. J. Schreiber, R. Bürger and D. I. Bolnick, The community effects of phenotypic and genetic variation within a predator population, Ecology, 92 (2011), 1582-1593.  doi: 10.1890/10-2071.1.

[49]

S. J. Schreiber and S. Patel, Evolutionarily induced alternative states and coexistence in systems with apparent competition, Natural Resource Modelling, 28 (2015), 475-496.  doi: 10.1111/nrm.12076.

[50]

H. L. Smith and X. Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete and Continuous Dynamical Systems–Series B, 1 (2001), 183-191.  doi: 10.3934/dcdsb.2001.1.183.

[51]

D. Tang, Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 4913-4928.  doi: 10.3934/dcdsb.2019037.

[52]

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

[53]

J. X. Velasco-Hernández, M. Núñez López, G. Ramírez-Santiago and M. Hernández-Rosales, On carrying-capacity construction, metapopulations and density-dependent mortality, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1099-1110.  doi: 10.3934/dcdsb.2017054.

[54]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.

[55]

H. Wang, K. Dunning, J. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete and Continuous Dynamical Systems–Series B, 12 (2009), 481-493.  doi: 10.3934/dcdsb.2009.12.481.

[56]

Y. Wu, N. Tuncer and M. Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1167-1187.  doi: 10.3934/dcdsb.2017057.

[57]

H. Yu, S. Zhong and R. P. Agarwal, Mathematics and dynamic analysis of an apparent competition community model with impulsive effect, Mathematical and Computer Modelling, 52 (2010), 25-36.  doi: 10.1016/j.mcm.2009.11.019.

Figure 1.  Stochastic persistence and the $ P^* $ rule for the host-parasitoid model (1). In A, there is $ k = 1 $ host species and the condition for stochastic persistence is met. In B, there are $ k = 4 $ host species which only differ in the variance of their $ R_i(t) $ terms. Parameter values: $ a_i(t) = 0.1 $ for all $ i $ and $ t $, $ R_i(t) = 0.9+1.1\beta^R_i(t) $ where $ \beta_i^R(t) $ are $ \beta $ distributed with both scale parameters $ = k+1-i $, and $ I(t) = 0.1+0.9\beta^I(t) $ where $ \beta^I(t) $ are $ \beta $ distributed with both scale parameters $ = 2 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[2]

Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332
[3]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697
[4]

John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843
[5]

Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060
[6]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216
[7]

Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092
[8]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[9]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[10]

Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102
[11]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[12]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[13]

Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa. Lyapunov exponents for random maps. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022058
[14]

Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640
[15]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[16]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
[17]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098
[18]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009
[19]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[20]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (239)
  • HTML views (116)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy