Alternative stable states and disease induced extinction

Dinesh Ekanayake^1, 2 ; Hunter La Croix³ ; Amy Ekanayake¹

¹ Department of Mathematics & Philosophy, Western Illinois University, 1 University Circle, Macomb, IL 61455, USA
² The Preston M. Green Department of Electrical & Systems Engineering, Washington University in St. Louis, 1 Brookings Drive, St. Louis, MO 63130, USA
³ The Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, 102G Crowley Hall, Notre Dame, IN 46556, USA

Mathematical modelling of natural phenomena, Tome 19 (2024), article no. 18

Résumé

In this research, we study a generalization of the SIR epidemic model to observe diseaseinduced extinction in the presence of alternative stable states. We model per capita reproduction and mortality rates as functions of both population density and external indicators reflecting temporal resource variations that impact stable states. We then obtain conditions to guarantee a unique global solution for the SIR model when the rate functions are discontinuous. We further obtain conditions for the stability of two states when the external indicators are assumed to be constant. We use both deterministic and stochastic epidemic simulations to analyze models with alternative stable states for real mammalian populations. Through numerical examples, we show that changes in external indicators can, in fact, lead to the collapse of a population subject to an epidemic. However, we find a high probability of species survival in the presence of environmental stochasticity, even when the corresponding deterministic models predict extinction of the host population.

Détail

DOI : 10.1051/mmnp/2024016

Bibliographie

[1] R.C. Lewontin The meaning of stability Diversity Stabil. Ecol. Syst. 1969 13 24

[2] B.E. Beisner, D.T. Haydon, K. Cuddington Alternative stable states in ecology Front. Ecol. Environ. 2003 376 382

[3] C. Folke, S. Carpenter, B. Walker, M. Scheffer, T. Elmqvist, L. Gunderson, C.S. Holling Regime shifts, resilience, and biodiversity in ecosystem management Annu. Rev. Ecol. Evol. Syst. 2004 557 581

[4] I.A. Van De Leemput, V. Dakos, M. Scheffer, E.H. Van Nes Slow recovery from local disturbances as an indicator for loss of ecosystem resilience Ecosystems 2018 141 152

[5] K.N. Suding, K.L. Gross, G.R. Houseman Alternative states and positive feedbacks in restoration ecology Trends Ecol. Evol. 2004 46 53

[6] F. De Castro, B. Bolker Mechanisms of disease-induced extinction Ecol. Lett. 2005 117 126

[7] K.E. Emmert, L.J.S. Allen Population extinction in deterministic and stochastic discrete-time epidemic models with periodic coefficients with applications to amphibian populations Nat. Resource Model. 2006 117 164

[8] H.R. Thieme, T. Dhirasakdanon, Z. Han, R. Trevino Species decline and extinction: synergy of infectious disease and allee effect? J. Biol. Dyn. 2009 305 323

[9] Y. Kang, C. Castillo-Chavez Multiscale analysis of compartment models with dispersal J. Biol. Dyn. 2012 50 79

[10] A. Friedman, A.-A. Yakubu Host demographic Allee effect, fatal disease, and migration: Persistence or extinction SIAM J. Appl. Math. 2012 1644 1666

[11] A.J. Ekanayake, D.B. Ekanayake A seasonal sir metapopulation model with an Allee effect with application to controlling plague in prairie dog colonies J. Biol. Dyn. 2015 262 290

[12] W.C. Allee, The Social Life of Animals. New York, W.W. Norton Company, Inc. (1938).

[13] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation. OUP, Oxford (2008).

[14] K.R. Hopper, R.T. Roush Mate finding, dispersal, number released, and the success of biological control introductions Ecol. Entomol. 1993 321 331

[15] H.G. Davis, C.M. Taylor, J.G. Lambrinos, D.R. Strong Pollen limitation causes an Allee effect in a wind- pollinated invasive grass (Spartina alterniflora) Proc. Natl. Acad. Sci. U.S.A. 2004 13804 13807

[16] K.-H. Choi, W.J. Kimmerer Mate limitation in an estuarine population of copepods Limnol. Oceanogr. 2008 1656

[17] M.S. Mooring, T.A. Fitzpatrick, T.T. Nishihira, D.D. Reisig Vigilance, predation risk, and the allee effect in desert bighorn sheep J. Wildlife Manag. 2004 519 532

[18] K.H. Elliott, G.S. Betini, D.R. Norris Fear creates an Allee effect: experimental evidence from seasonal populations Proc. Roy. Soc. B: Biol. Sci. 2017 20170878

[19] S.A. West, A.S. Griffin, A. Gardner Social semantics: altruism, cooperation, mutualism, strong reciprocity and group selection J. Evol. Biol. 2007 415 432

[20] P.A. Stephens, W.J. Sutherland Consequences of the Allee effect for behaviour, ecology and conservation Trends Ecol. Evol. 1999 401 405

[21] E. Angulo, G.M. Luque, S.D. Gregory, J.W. Wenzel, C. Bessa-Gomes, L. Berec, F. Courchamp Allee effects in social species J. Anim. Ecol. 2018 47 58

[22] J.-M. Gaillard, M. Festa-Bianchet, N.G. Yoccoz Population dynamics of large herbivores: variable recruitment with constant adult survival Trends Ecol. Evol. 1998 58 63

[23] J.M. Gaillard, M. Festa-Bianchet, N.G. Yoccoz, A. Loison, C. Toigo Temporal variation in fitness components and population dynamics of large herbivores Annu. Rev. Ecol. Syst. 2000 367 393

[24] L.L. Eberhardt A paradigm for population analysis of long-lived vertebrates Ecology 2002 2841 2854

[25] C. Bonenfant, J.-M. Gaillard, T. Coulson, M. Festa-Bianchet, A. Loison, M. Garel, L.E. Loe, P. Blanchard, N. Pettorelli, N. Owen-Smith Empirical evidence of density-dependence in populations of large herbivores Adv. Ecol. Res. 2009 313 357

[26] K.L. Monteith, V.C. Bleich, T.R. Stephenson, B.M. Pierce, M.M. Conner, J.G. Kie, R.T. Bowyer Life-history characteristics of mule deer: effects of nutrition in a variable environment Wildlife Monogr. 2014 1 62

[27] E.J. Chapman, C.J. Byron The flexible application of carrying capacity in ecology Global Ecol. Conserv. 2018 e00365

[28] D.B. Ekanayake, A.J. Ekanayake, J. Hunt, C.L. Miller-Hunt Modeling reproduction of whitetail deer and its applications J. Theor. Biol. 2018 90 102

[29] F. Valladares, D. Sanchez-Gomez, M.A. Zavala Quantitative estimation of phenotypic plasticity: bridging the gap between the evolutionary concept and its ecological applications J. Ecol. 2006 1103 1116

[30] E.E. Werner, S.D. Peacor A review of trait-mediated indirect interactions in ecological communities Ecology 2003 1083 1100

[31] R. Langvatn, S.D. Albon, T. Burkey, T.H. Clutton-Brock Climate, plant phenology and variation in age of first reproduction in a temperate herbivore J. Anim. Ecol. 1996 653 670

[32] N. Pettorelli, J.-M. Gaillard, G. Van Laere, P. Duncan, P. Kjellander, O. Liberg, D. Delorme, D. Maillard Variations in adult body mass in roe deer: the effects of population density at birth and of habitat quality Proc. Roy. Soc. Lond. Ser. B: Biol. Sci. 2002 747 753

[33] B.-E. Sæther, M. Heim Ecological correlates of individual variation in age at maturity in female moose (Alces alces): the effects of environmental variability J. Anim. Ecol. 1993 482 489

[34] D.R. Edmunds, M.J. Kauffman, B.A. Schumaker, F.G. Lindzey, W.E. Cook, T.J. Kreeger, R.G. Grogan, T.E Cornish Chronic wasting disease drives population decline of white-tailed deer PLoS One 2016 e0161127

[35] R.J. Monello, J.G. Powers, N.T. Hobbs, T.R. Spraker, M.K. Watry, M.A. Wild Survival and population growth of a free-ranging elk population with a long history of exposure to chronic wasting disease J. Wildlife Manag. 2014 214 223

[36] D.E. Brown, D. Warnecke, T. Mckinney Effects of midsummer drought on mortality of doe pronghorn (Antilocapra americana) Southwest. Natural. 2006 220 225

[37] D.C. Simpson, L.A. Harveson, C.E. Brewer, R.E. Walser, A.R. Sides Influence of precipitation on pronghorn demography in Texas J. Wildlife Manag. 2007 906 910

[38] P.F. Jones, A.F. Jakes, D.R. Eacker, M. Hebblewhite Annual pronghorn survival of a partially migratory population J. Wildlife Manag. 2020 1114 1126

[39] L. Cai, G. Chen, D. Xiao Multiparametric bifurcations of an epidemiological model with strong Allee effect J. Math. Biol. 2013 185 215

[40] Y. Liu, S. Ruan, L. Yang Stability transition of persistence and extinction in an avian influenza model with Allee effect and stochasticity Commun. Nonlinear Sci. Numer. Simul. 2020 105416

[41] R. Lande Genetics and demography in biological conservation Science 1988 1455 1460

[42] J.K. Hale, Ordinary Differential Equations. Dover (1980).

[43] J.M. Heffernan, R.J. Smith, L.M. Wahl Perspectives on the basic reproductive ratio J. Roy. Soc. Interface 2005 281 293

[44] D. Serre, Matrices: Theory and Applications. Springer (2010).

[45] E. Allen, editor, Modeling with Itô Stochastic Differential Equations. Mathematical Modelling: Theory and Applications. Springer (2007).

[46] F.M. Hilker, M. Langlais, H. Malchow The Allee effect and infectious diseases: extinction, multistability, and the (dis-) appearance of oscillations Am. Natural. 2009 72 88

[47] A. Friedman, A.-A. Yakubu Fatal disease and demographic Allee effect: population persistence and extinction J. Biol. Dyn. 2012 495 508

[48] J.F. Cully, E.S. Williams Interspecific comparisons of sylvatic plague in prairie dogs J. Mammal. 2001 894 905

[49] T.M. Blackburn, K.J. Gaston, N. Loder Geographic gradients in body size: a clarification of Bergmann’s rule Diversity Distrib. 1999 165 174

[50] A.N. Hamir, R.A. Kunkle, E.M. Nicholson, J.M. Miller, S.M. Hall, H. Schoenenbruecher, B.W. Brunelle, J.A. Richt Preliminary observations on the experimental transmission of chronic wasting disease (CWD) from elk and white-tailed deer to fallow deer J. Compar. Pathol. 2008 121 130

[51] J.D. Hoffman, H.H. Genoways, R.R. Jones Factors influencing long-term population dynamics of pronghorn (Antilocapra americana): evidence of an Allee effect J. Mammal. 2010 1124 1134

[52] B. Grzimek, Grzimek’s Encyclopedia of Mammals. McGraw-Hill (1990).