February  2019, 24(2): 737-754. doi: 10.3934/dcdsb.2018205

Dirac-concentrations in an integro-pde model from evolutionary game theory

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

Received  August 2017 Revised  February 2018 Published  June 2018

Fund Project: The first author is supported by NSF grant DMS-1411476.

Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Motivated by the existence of moving Dirac-concentrations in the time-dependent problem, we study the qualitative properties of steady states in the limit of small diffusion. Under different conditions on the growth rate and interaction kernel as motivated by the framework of adaptive dynamics, we will show that as the diffusion rate tends to zero the steady state concentrates (ⅰ) at a single location; (ⅱ) at two locations simultaneously; or (ⅲ) at one of two alternative locations. The third result in particular shows that solutions need not be unique. This marks an important difference of the non-local equation with its local counterpart.

Citation: King-Yeung Lam. Dirac-concentrations in an integro-pde model from evolutionary game theory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 737-754. doi: 10.3934/dcdsb.2018205
References:
[1]

A. S. AcklehJ. Cleveland and H. R. Thieme, Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces, J. Differential Equations, 261 (2016), 1472-1505.  doi: 10.1016/j.jde.2016.04.008.  Google Scholar

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G. Barles, An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, 49-109, Lecture Notes in Math., 2074, Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36433-4_2.  Google Scholar

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N. ChampagnatR. Ferrière and S. et Mèlèard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theor. Popul. Biol., 69 (2006), 297-321.  doi: 10.1016/j.tpb.2005.10.004.  Google Scholar

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R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics, Theor. Pop. Biol., 67 (2005), 47-59.  doi: 10.1016/j.tpb.2004.08.001.  Google Scholar

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L. DesvillettesP.-E. JabinS. Mischler and G. Raoul, On mutation-selection dynamics, Commun. Math. Sci., 6 (2008), 729-747.  doi: 10.4310/CMS.2008.v6.n3.a10.  Google Scholar

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O. Diekmann, A beginner's guide to adaptive dynamics, Banach Center Publications, 63 (2004), 47-86.   Google Scholar

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O. DiekmannP.-E. JabinS. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Pop. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.  Google Scholar

[9]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar

[10]

R. A. Fisher, The Genetical Theory of Natural Selection, Oxford University Press, Oxford, 1999.  Google Scholar

[11]

W. HaoK.-Y. Lam and Y. Lou, Concentration phenomena in an integro-PDE model for evolution of conditional dispersal, Indiana Univ. Math. J., 272 (2017), 1755-1790.  doi: 10.1016/j.jfa.2016.11.017.  Google Scholar

[12]

S. Gandon and S. Mirrahimi, A Hamilton-Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations, Comptes Rendus Mathematique, 355 (2016), 155-160.  doi: 10.1016/j.crma.2016.12.001.  Google Scholar

[13]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[14]

J. Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations, 226 (2006), 541-557.  doi: 10.1016/j.jde.2006.02.008.  Google Scholar

[15]

S. F. Iglesias and S. Mirrahimi, Long time evolutionary dynamics of phenotypically structured populations in time periodic environments, arXiv: 1803.03547 [math. AP]. Google Scholar

[16]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci. USA, 54 (1965), 731-736.  doi: 10.1073/pnas.54.3.731.  Google Scholar

[17]

K.-Y. Lam and Y. Lou, Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal, Bull. Math. Biol., 76 (2014), 261-291.  doi: 10.1007/s11538-013-9901-y.  Google Scholar

[18]

K.-Y. Lam and Y. Lou, An integro-PDE model for evolution of random dispersal, J. Funct. Anal., 272 (2017), 1755-1790.  doi: 10.1016/j.jfa.2016.11.017.  Google Scholar

[19]

K. -Y. Lam, Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal, Calc. Var. Partial Differential Equations, 56 (2017), Art. 79, 32pp. doi: 10.1007/s00526-017-1157-1.  Google Scholar

[20]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[21]

A. LorzS. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Diff. Equations, 36 (2011), 1071-1098.  doi: 10.1080/03605302.2010.538784.  Google Scholar

[22]

P. Magal and G. F. Webb, Mutation, selection, and recombination in a model of phenotype evolution, Discrete Cont. Dynam. Syst., 6 (2000), 221-236.   Google Scholar

[23]

G. MeszènaM. GyllenbergF. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of darwinian evolution, Phys. Rev. Lett., 95 (2005), 78-105.   Google Scholar

[24]

B. Perthame and P. E. Souganidis, Rare mutations limit of a steady state dispersal evolution model, Math. Model. Nat. Phenom., 11 (2016), 154-166.  doi: 10.1051/mmnp/201611411.  Google Scholar

[25]

M. V. Safonov and N. V. Krylov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175.   Google Scholar

[26]

A. Sasaki, Clumped distribution by neighborhood competition, J. Theor. Biol., 186 (1997), 415-430.   Google Scholar

[27]

H. Smith and H. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[28]

L. SunJ. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9.  Google Scholar

[29]

J. WickmanS. DiehlC. A. KausmeierA. B. Ryabov and A. Brännström, Determining selection across heterogeneous landscapes: A perturbation-based method and its application to modeling evolution in space, Am. Nat., 189 (2017), 381-395.  doi: 10.1086/690908.  Google Scholar

show all references

References:
[1]

A. S. AcklehJ. Cleveland and H. R. Thieme, Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces, J. Differential Equations, 261 (2016), 1472-1505.  doi: 10.1016/j.jde.2016.04.008.  Google Scholar

[2]

G. Barles, An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, 49-109, Lecture Notes in Math., 2074, Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36433-4_2.  Google Scholar

[3]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.  doi: 10.1007/BF01215194.  Google Scholar

[4]

N. ChampagnatR. Ferrière and S. et Mèlèard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theor. Popul. Biol., 69 (2006), 297-321.  doi: 10.1016/j.tpb.2005.10.004.  Google Scholar

[5]

R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics, Theor. Pop. Biol., 67 (2005), 47-59.  doi: 10.1016/j.tpb.2004.08.001.  Google Scholar

[6]

L. DesvillettesP.-E. JabinS. Mischler and G. Raoul, On mutation-selection dynamics, Commun. Math. Sci., 6 (2008), 729-747.  doi: 10.4310/CMS.2008.v6.n3.a10.  Google Scholar

[7]

O. Diekmann, A beginner's guide to adaptive dynamics, Banach Center Publications, 63 (2004), 47-86.   Google Scholar

[8]

O. DiekmannP.-E. JabinS. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Pop. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.  Google Scholar

[9]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar

[10]

R. A. Fisher, The Genetical Theory of Natural Selection, Oxford University Press, Oxford, 1999.  Google Scholar

[11]

W. HaoK.-Y. Lam and Y. Lou, Concentration phenomena in an integro-PDE model for evolution of conditional dispersal, Indiana Univ. Math. J., 272 (2017), 1755-1790.  doi: 10.1016/j.jfa.2016.11.017.  Google Scholar

[12]

S. Gandon and S. Mirrahimi, A Hamilton-Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations, Comptes Rendus Mathematique, 355 (2016), 155-160.  doi: 10.1016/j.crma.2016.12.001.  Google Scholar

[13]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[14]

J. Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations, 226 (2006), 541-557.  doi: 10.1016/j.jde.2006.02.008.  Google Scholar

[15]

S. F. Iglesias and S. Mirrahimi, Long time evolutionary dynamics of phenotypically structured populations in time periodic environments, arXiv: 1803.03547 [math. AP]. Google Scholar

[16]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci. USA, 54 (1965), 731-736.  doi: 10.1073/pnas.54.3.731.  Google Scholar

[17]

K.-Y. Lam and Y. Lou, Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal, Bull. Math. Biol., 76 (2014), 261-291.  doi: 10.1007/s11538-013-9901-y.  Google Scholar

[18]

K.-Y. Lam and Y. Lou, An integro-PDE model for evolution of random dispersal, J. Funct. Anal., 272 (2017), 1755-1790.  doi: 10.1016/j.jfa.2016.11.017.  Google Scholar

[19]

K. -Y. Lam, Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal, Calc. Var. Partial Differential Equations, 56 (2017), Art. 79, 32pp. doi: 10.1007/s00526-017-1157-1.  Google Scholar

[20]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[21]

A. LorzS. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Diff. Equations, 36 (2011), 1071-1098.  doi: 10.1080/03605302.2010.538784.  Google Scholar

[22]

P. Magal and G. F. Webb, Mutation, selection, and recombination in a model of phenotype evolution, Discrete Cont. Dynam. Syst., 6 (2000), 221-236.   Google Scholar

[23]

G. MeszènaM. GyllenbergF. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of darwinian evolution, Phys. Rev. Lett., 95 (2005), 78-105.   Google Scholar

[24]

B. Perthame and P. E. Souganidis, Rare mutations limit of a steady state dispersal evolution model, Math. Model. Nat. Phenom., 11 (2016), 154-166.  doi: 10.1051/mmnp/201611411.  Google Scholar

[25]

M. V. Safonov and N. V. Krylov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175.   Google Scholar

[26]

A. Sasaki, Clumped distribution by neighborhood competition, J. Theor. Biol., 186 (1997), 415-430.   Google Scholar

[27]

H. Smith and H. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[28]

L. SunJ. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9.  Google Scholar

[29]

J. WickmanS. DiehlC. A. KausmeierA. B. Ryabov and A. Brännström, Determining selection across heterogeneous landscapes: A perturbation-based method and its application to modeling evolution in space, Am. Nat., 189 (2017), 381-395.  doi: 10.1086/690908.  Google Scholar

Figure 1.  The left, center and right panels illustrate the sign of $\frac{K(x,y)}{r(x)} - \frac{K(y,y)}{r(y)}$ as a function of $x$ and $y$, under the assumptions of Theorems 1, 2 and 3 respectively. Here $x$ and $y$ are the strategy of the invader and resident species respectively. $\frac{K(x,y)}{r(x)} - \frac{K(y,y)}{r(y)} < 0$ (resp. $>0$) means invasion of resident with strategy ''$y$" by invader with strategy ''$x$" is a success (resp. failure)
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