American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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

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

[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.  Google Scholar

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

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

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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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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. Google Scholar

[12]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

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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 $
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