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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[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.

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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