Research article

Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence

  • Received: 30 August 2019 Accepted: 17 November 2019 Published: 11 December 2019
  • The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect. In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density. The habitat patch is one-dimensional and is surrounded by a tuneable hostile matrix. We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies. Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them. Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch. Our results are obtained mathematically through the method of subsuper solutions, time map analysis, and numerical computations using Wolfram Mathematica.

    Citation: James T. Cronin, Nalin Fonseka, Jerome Goddard II, Jackson Leonard, Ratnasingham Shivaji. Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1718-1742. doi: 10.3934/mbe.2020090

    Related Papers:

  • The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect. In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density. The habitat patch is one-dimensional and is surrounded by a tuneable hostile matrix. We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies. Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them. Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch. Our results are obtained mathematically through the method of subsuper solutions, time map analysis, and numerical computations using Wolfram Mathematica.


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    [1] W. C. Allee, Animal Aggregations; a Study in General Sociology, University of Chicago Press, Chicago, 1931.
    [2] N. Knowlton, Thresholds and multiple stable states in coral reef community dynamics, Integr. Comp. Biol., 32 (1992), 674-682.
    [3] B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, Nat. Resour. Modell., 3 (1989), 481-538.
    [4] M. A. Lewis, P. Kareiva, Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 43 (1993), 141-158.
    [5] M. Fischer, M. Hock, M. Paschke, Low genetic variation reduces cross-compatibility and offspring fitness in populations of a narrow endemic plant with a self-incompatibility system, Conserv. Genet., 4 (2003), 325-336.
    [6] F. Courchamp, L. Berec, J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008.
    [7] A. M. Kramer, B. Dennis, A. M. Liebhold, J. M. Drake, The evidence for allee effects, Popul. Ecol., 51 (2009), 341-354.
    [8] R. M. Sibly, D. Barker, M. C. Denham, J. Hone, M. Pagel, On the regulation of populations of mammals, birds, fish, and insects, Science, 309 (2005), 607-610.
    [9] J. A. Hutchings, Thresholds for impaired species recovery, Proc. R. Soc. B: Biol. Sci., 282 (2015), 1-11.
    [10] C. M. Taylor, A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895-908.
    [11] A. K. Shaw, H. Kokko, M. G. Neubert, Sex difference and allee effects shape the dynamics of sex-structured invasions, J. Animal Ecol., 87 (2018), 36-46.
    [12] D. M. Johnson, A. M. Liebhold, P. C. Tobin, O. N. Bjrnstad, Allee effects and pulsed invasion by the gypsy moth, Nature, 444 (2006), 361-363.
    [13] P. C. Tobin, L. Berec, A. M. Liebhold, Exploiting allee effects for managing biological invasions, Ecol. Lett., 14 (2011), 615-624.
    [14] J. C. Blackwood, L. Berec, T. Yamanaka, R. S. Epanchin-Niell, A. Hastings, A. M. Liebhold, Bioeconomic synergy between tactics for insect eradication in the presence of allee effects, Proc. R. Soc. B: Biol. Sci., 279 (2012), 2807-2815.
    [15] R. R. Regoes, D. Ebert, S. Bonhoeffer, Dose-dependent infection rates of parasites produce the allee effect in epidemiology, Proc. R. Soc. London. Ser. B: Biol. Sci., 269 (2002), 271-279.
    [16] A. Deredec, F. Courchamp, Combined impacts of allee effects and parasitism, Oikos, 112 (2006), 667-679.
    [17] F. Hilker, M. Langlais, H. Malchow, The allee effect and infectious diseases: Extinction, multistability, and the (dis)appearance of oscillations, Am. Nat., 173 (2009), 72-88.
    [18] K. S. Korolev, J. B. Xavier, J. Gore, Turning ecology and evolution against cancer, Nat. Rev. Cancer, 14 (2014), 1-10.
    [19] L. Sewalt, K. Harley, P. van Heijster, S. Balasuriya, Influences of allee effects in the spreading of malignant tumours, J. Theor. Biol., 394 (2016), 77-92.
    [20] M. A. Pires, S. M. Duarte-Queirs, Optimal dispersal in ecological dynamics with allee effect in metapopulations, PLoS One, 14 (2019), 1-15.
    [21] C. E. Brassil, Mean time to extinction of a metapopulation with an allee effect, Ecol. Modell., 143 (2001), 9-16.
    [22] P. Amarasekare, Allee effects in metapopulation dynamics, Am. Nat., 152 (1998), 298-302.
    [23] S. R. Zhou, G. Wang, Allee-like effects in metapopulation dynamics, Math. Biosci., 189 (2004), 103-113.
    [24] S. Petrovskii, A. Morozov, B. L. Li, Regimes of biological invasion in a predator-prey system with the allee effect, Bull. Math. Biol., 67 (2005), 637-661.
    [25] I. D. Jonsen, R. S. Bourchier, J. Roland, Influence of dispersal, stochasticity, and an allee effect on the persistence of weed biocontrol introductions, Ecol. Modell., 203 (2007), 521-526.
    [26] R. R. Veit, M. A. Lewis, Dispersal, population growth, and the allee effect: Dynamics of the house finch invasion of eastern north america, Am. Nat., 148 (1996), 255-274.
    [27] O. Kindvall, K. Vessby, S. Berggren, G. Hartman, Individual mobility prevents an allee effect in sparse populations of the bush cricket metrioptera roeseli: An experimental study, Oikos, 81 (1998), 449-457.
    [28] D. Bonte, L. Lens, J. P. Maelfait, Lack of homeward orientation and increased mobility result in high emigration rates from low-quality fragments in a dune wolf spider, J. Anim. Ecol., 73 (2004), 643-650.
    [29] P. Amarasekare, The role of density-dependent dispersal in sourcesink dynamics, J. Theor. Biol., 226 (2004), 159-168.
    [30] D. E. Bowler, T. G. Benton, Causes and consequences of animal dispersal strategies: Relating individual behaviour to spatial dynamics, Biol. Rev., 80 (2005), 205-225.
    [31] E. Matthysen, Multicausality of Dispersal: A Review, Oxford University Press, United Kingdom, 2012, 3-18.
    [32] R. Harman, J. Goddard, R. Shivaji, J. T. Cronin, Frequency of cccurrence and population-dynamic consequences of different forms of density-dependent emigration, Am. Nat., Forthcoming.
    [33] R. S. Cantrell, C. Cosner, Density dependent behavior at habitat boundaries and the allee effect, Bull. Math. Biol., 69 (2007), 2339-2360.
    [34] J. Goddard II, Q. Morris, C. Payne, R. Shivaji, A diffusive logistic equation with u-shaped density dependent dispersal on the boundary, Topol. Methods Nonlinear Anal., 53 (2019), 335-349.
    [35] J. Drake, A. Kramer, Allee effects, Nat. Educ. Knowl., 3 (2011), 2.
    [36] J. Shi, R. Shivaji, Persistence in reaction diffusion models with weak allee effect, J. Math. Biol., 52 (2006), 807-829.
    [37] S. A. Levin, Dispersion and population interactions, Am. Nat., 108 (1974), 207-228.
    [38] S. A. Levin, The role of theoretical ecology in the description and understanding of populations in heterogeneous environments, Am. Zool., 21 (1981), 865-875.
    [39] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, 1979.
    [40] A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Springer, Berlin, 1980.
    [41] J. D. Murray, Mathematical Biology. II, 3rd edition, Springer-Verlag, New York, 2003.
    [42] R. S. Cantrell, C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2003.
    [43] E. E. Holmes, M. A. Lewis, R. R. V. Banks, Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75 (1994), 17-29.
    [44] J. T. Cronin, J. Goddard II, R. Shivaji, Effects of patch matrix-composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933-3975.
    [45] R. S. Cantrell, C. Cosner, On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains, J. Differ. Eq., 231 (2006), 768-804.
    [46] N. Foneska, J. Goddard II, Q. Morris, R. Shivaji, B. Son, On the effects of the exterior matrix hostility and a u-shaped density dependent dispersal on a diffusive logistic growth model, Discrete Contin. Dyn. Syst. Ser. B, Forthcoming.
    [47] J. Goddard II, R. Shivaji, Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions, Proc. R. Soc. Edinburgh, 147 (2017), 1019-1040.
    [48] S. Robinson, M. A. Rivas, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var. 25 (2019), 45-69.
    [49] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
    [50] J. Goddard II, Q. Morris, S. Robinson, R. Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Boundary Value Probl., 170 (2018), 1-17.
    [51] P. A. Stephens, W. J. Sutherland, R. P. Freckleton, What is the allee effect?, Oikos, 87 (1999), 185-190.
    [52] I. Hanski, Metapopulation Ecology, Oxford University Press, Oxford, 1999.
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