On the Allee effect and directed movement on the whole space

Chris Cosner; Nancy Rodríguez; Chris Cosner; Nancy Rodríguez

doi:10.3934/mbe.2023347

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8010-8030. doi: 10.3934/mbe.2023347

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On the Allee effect and directed movement on the whole space

Chris Cosner ¹,
Nancy Rodríguez ^{2
,
,}

1.
University of Miami, Department of Mathematics, 1365 Memorial Drive, Ungar 515, Coral Gables, FL 33146, USA
2.
CU Boulder, Department of Applied Mathematics, 11 Engineering Dr, Boulder, CO 80309, USA

Academic Editor: Zhi-An Wang

Received: 18 October 2022 Revised: 03 January 2023 Accepted: 08 January 2023 Published: 23 February 2023

It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.
- reaction-advection-diffusion equations,
- Allee effect,
- directed movement,
- evolutionary game theory
Citation: Chris Cosner, Nancy Rodríguez. On the Allee effect and directed movement on the whole space[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8010-8030. doi: 10.3934/mbe.2023347

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Abstract

It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.

References

[1]	W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, 1931.
[2]	V. Krivan, The Allee-type ideal free distribution, J. Math. Biol., 69 (2013), 1497–1513. https://doi.org/10.1007/s00285-013-0742-y doi: 10.1007/s00285-013-0742-y
[3]	W. C. Allee, E. S. Bowen, Studies in animal aggregations: Mass protection against colloidal silver among goldfishes, J. Exp. Zool., 61 (1932), 185–207. https://doi.org/10.1002/jez.1400610202 doi: 10.1002/jez.1400610202
[4]	G. Livadiotis, L. Assas, S. Elaydi, E. Kwessi, D. Ribble, Competition models with Allee effects, 20 (2014), 1127–1151. http://dx.doi.org/10.1080/10236198.2014.897341
[5]	E. Angulo, G. M. Luque, S. D. Gregory, J. W. Wenzel, C. Bessa-Gomes, L. Berec, et al., Allee effects in social species, J. Anim. Ecol., 87 (2018), 47–58.
[6]	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., 9 (2015), 262–290. https://doi.org/10.1080/17513758.2014.978400 doi: 10.1080/17513758.2014.978400
[7]	G. Sempo, S. Canonge, J. L. Deneubourg, From aggregation to dispersion: How habitat fragmentation prevents the emergence of consensual decision making in a group, PLoS ONE, 8 (2013). https://doi.org/10.1371/journal.pone.0078951 doi: 10.1371/journal.pone.0078951
[8]	R. Cantrell, C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley & Sons, West Sussex, 2003.
[9]	A. P. Ramakrishnan, Dispersal-Migration, in Encyclopedia of Ecology, (2008), 185–191.
[10]	J. Bedrossian, N. Rodríguez, Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in Rd, DCDS-B, 19 (2014), 1–33. https://doi.org/10.3934/dcdsb.2014.19.1279 doi: 10.3934/dcdsb.2014.19.1279
[11]	A. Bertozzi, D. Slepcev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure Appl. Anal., 9 (2010), 1617–1637. https://doi.org/10.3934/cpaa.2010.9.1617 doi: 10.3934/cpaa.2010.9.1617
[12]	G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272–296. https://doi.org/10.1137/16M107245 doi: 10.1137/16M107245
[13]	D. Li, X. Zhang, On a nonlocal aggregation model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 27 (2010), 301–323. https://doi.org/10.3934/dcds.2010.27.301 doi: 10.3934/dcds.2010.27.301
[14]	J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Appl. Math. Lett., 24 (2011), 1927–1932. https://doi.org/10.1016/j.aml.2011.05.022 doi: 10.1016/j.aml.2011.05.022
[15]	C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst.-Ser. A, 34 (2014), 1701–1745. https://doi.org/10.3934/dcds.2014.34.1701 doi: 10.3934/dcds.2014.34.1701
[16]	R. S. Cantrell, C. Cosner, Conditional persistence in logistic models via nonlinear diffusion, Proc. R. Soc. Edinburgh Sect. A: Math., 132 (2002), 267–281. https://doi.org/10.1017/S0308210500001621 doi: 10.1017/S0308210500001621
[17]	A. Astudillo Fernandez, T. Hance, J. L. Deneubourg, Interplay between Allee effects and collective movement in metapopulations, Oikos, 121 (2012), 813–822. https://doi.org/10.1111/j.1600-0706.2011.20181.x doi: 10.1111/j.1600-0706.2011.20181.x
[18]	D. C. Speirs, W. Gurney, Population persistence in rivers and estuaries, Ecol. Soc. Am., 82 (2001), 1219–1237. https://doi.org/10.1890/0012-9658(2001)082[1219:PPIRAE]2.0.CO;2 doi: 10.1890/0012-9658(2001)082[1219:PPIRAE]2.0.CO;2
[19]	Y. Jin, M. A. Lewis, Seasonal influences on population spread and persistence in streams: Spreading speeds, J. Math. Biol., 65 (2011), 403–439.
[20]	F. Lutscher, M. A. Lewis, E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129–2160. https://doi.org/10.1007/s11538-006-9100-1 doi: 10.1007/s11538-006-9100-1
[21]	F. Lutscher, R. M. Nisbet, E. Pachepsky, Population persistence in the face of advection, Theor. Ecol., 3 (2010), 271–284. https://doi.org/10.1007/s12080-009-0068-y doi: 10.1007/s12080-009-0068-y
[22]	F. Lutscher, E. Pachepsky, M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749–772. https://doi.org/10.1137/05063615 doi: 10.1137/05063615
[23]	Y. Wang, J. Shi, J. Wang, Persistence and extinction of population in reaction-diffusion-advection model with strong Allee effect growth, J. Math. Biol., 78 (2019), 2093–2140. https://doi.org/10.1007/s00285-019-01334-7 doi: 10.1007/s00285-019-01334-7
[24]	F. Belgacem, C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Can. Appl. Math. Q., 3 (1995), 379–397.
[25]	C. Cosner, Y. Lou, Does movement toward better environments always benefit a population, J. Math. Anal. Appl., 277 (2003), 489–503. https://doi.org/10.1016/S0022-247X(02)00575-9 doi: 10.1016/S0022-247X(02)00575-9
[26]	P. Grindrod, Models of individual aggregation or clustering in single and multi-species communities, J. Math. Biol., 26 (1988), 651–660. https://doi.org/10.1007/BF00276146 doi: 10.1007/BF00276146
[27]	V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Am. Math. Soc., 356 (2004), 2739–2756.
[28]	S. M. Flaxman, Y. Lou, Tracking prey or tracking the prey's resource? Mechanisms of movement and optimal habitat selection by predators, J. Theor. Biol., 256 (2009), 187–200. https://doi.org/10.1016/j.jtbi.2008.09.024 doi: 10.1016/j.jtbi.2008.09.024
[29]	C. Cosner, M. Winkler, Well-posedness and qualitative properties of a dynamical model for the ideal free distribution, J. Math. Biol., 69 (2013), 1343–1382. https://doi.org/10.1007/s00285-013-0733-z doi: 10.1007/s00285-013-0733-z
[30]	Y. Lou, F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319–1342. https://doi.org/10.1007/s00285-013-0730-2 doi: 10.1007/s00285-013-0730-2
[31]	C. Cosner, N. Rodriguez, The effect of directed movement on the strong Allee effect, SIAM J. Appl. Math., 81 (2021), 407–433. https://doi.org/10.1137/20M1330178 doi: 10.1137/20M1330178
[32]	M. Krzyżański, On the solutions of equations of the parabolic type determined in an unlimited region, Bull. Am. Math. Soc., 47 (1941), 911–915.
[33]	X. Chen, R. Hambrock, Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361–386. https://doi.org/10.1007/s00285-008-0166-2 doi: 10.1007/s00285-008-0166-2
[34]	J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196–218. https://doi.org/10.1093/biomet/38.1-2.196 doi: 10.1093/biomet/38.1-2.196
[35]	V. Volpert, S. Petrovskii, Reaction–diffusion waves in biology, Phys. Life Rev., 6 (2009), 267–310. http://dx.doi.org/10.1016/j.plrev.2009.10.002 doi: 10.1016/j.plrev.2009.10.002
[36]	A. J. Perumpanani, J. A. Sherratt, J. Norbury, H. M. Byrne, A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion, Physica D, 126 (1999), 145–159. https://doi.org/10.1016/S0167-2789(98)00272-3 doi: 10.1016/S0167-2789(98)00272-3
[37]	P. W. Bates, P. Fife, X. Ren, X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105–136. https://doi.org/10.1007/s002050050037 doi: 10.1007/s002050050037
[38]	D. G. Aronson, H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, Springer, Berlin, (1975), 5–49. https://doi.org/10.1007/BFb0070595
[39]	A. Zlatos, Sharp transition between extinction and propagation of reaction, J. Am. Math. Soc., 19 (2005), 251–263.
[40]	M. Schienbein, K. Franke, H. Gruler, Random walk and directed movement: comparison between inert particles and self-organized molecular machines, Phys. Rev. E, 49 (1994), 5462–5472. https://doi.org/10.1103/PhysRevE.49.5462 doi: 10.1103/PhysRevE.49.5462
[41]	G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ., 1996.
[42]	C. Cosner, Asymptotic behavior of solutions of second order parabolic partial differential equations with unbounded coefficients, J. Differ. Equations, 35 (1980), 407–428. https://doi.org/10.1016/0022-0396(80)90036-4 doi: 10.1016/0022-0396(80)90036-4
[43]	P. Besela, On solutions of Fourier's first problem for a system of non-linear equations in an unbounded domain, Ann. Pol. Math., 13 (1963), 247–265.
[44]	D. G. Aronson, H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33–76.
[45]	G. E. Uhlenbeck, L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev., 36 (1930), 823–841. https://doi.org/10.1103/PhysRev.36.823 doi: 10.1103/PhysRev.36.823
[46]	T. Björk, Arbitrage Theory in Continuous Time, Oxford University Press, 2009.
[47]	C. L. MacLeod, Ž. Ivezić, C. S. Kochanek, S. Kozłlowski, B. Kelly, E. Bullock, et al., Modeling the time variability of SDSS Stripe 82 quasars as a damped random walk, Astrophys. J., 721 (2010), 1014–1033. https://doi.org/10.1088/0004-637X/721/2/1014 doi: 10.1088/0004-637X/721/2/1014
[48]	E. P. Martins, Estimating the rate of phenotypic evolution from comparative data, Am. Nat., 144 (1994), 193–209.
[49]	R. S. Cantrell, C. Cosner, Y. Lou, Advection-mediated coexistence of competing species, Proc. R. Soc. Edinburgh Sec. A: Math., 137 (2007), 497–518. https://doi.org/10.1017/S0308210506000047 doi: 10.1017/S0308210506000047
[50]	K. Y. Lam, W. M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051–1067. https://doi.org/10.3934/dcds.2010.28.1051 doi: 10.3934/dcds.2010.28.1051
[51]	K. J. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differ. Equations, 250 (2011), 161–181. https://doi.org/10.1016/j.jde.2010.08.028 doi: 10.1016/j.jde.2010.08.028
[52]	V. Giunta, T. Hillen, M. Lewis, J. R. Potts, Local and global existence for nonlocal multispecies advection-diffusion models, SIAM J. Appl. Dyn. Syst., 21 (2022), 1686–1708. https://doi.org/10.1137/21M1425992 doi: 10.1137/21M1425992
[53]	J. R. Potts, M. A. Lewis, How memory of direct animal interactions can lead to territorial pattern formation, J. R. Soc. Interface, 13 (2016). https://doi.org/10.1098/rsif.2016.0059 doi: 10.1098/rsif.2016.0059
[54]	N. P. Taylor, H. Kim, A. L. Krause, R. A. Van Gorder, A non-local cross-diffusion model of population dynamics Ⅰ: Emergent spatial and spatiotemporal patterns, Bull. Math. Biol., 82 (2020). https://doi.org/10.1007/s11538-020-00786-z doi: 10.1007/s11538-020-00786-z

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