Evolutionarily Stable Dispersal Strategies in Heterogeneous Environments

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1 Evolutionarily Stable Dispersal Strategies in Heterogeneous Environments Yuan Lou Department of Mathematics Mathematical Biosciences Institute Ohio State University Columbus, OH 43210, USA Yuan Lou (Ohio State) NC State / 43

2 Talk Outline 1 Unbiased dispersal 2 Balanced dispersal 3 Biased dispersal 4 Fitness-dependent dispersal Yuan Lou (Ohio State) NC State / 43

3 Evolution of Dispersal Unbiased dispersal How should organisms move optimally" in heterogeneous environments? Yuan Lou (Ohio State) NC State / 43

4 Previous works Unbiased dispersal Levin 76; Hastings 83; Holt 85; McPeek and Holt 92; Holt and McPeek 1996; Dockery et al. 1998; Kirkland et al. 2006; Abrams 2007; Armsworth and Roughgarden 2008; Amarasekare 2010 Yuan Lou (Ohio State) NC State / 43

5 Previous works Unbiased dispersal Levin 76; Hastings 83; Holt 85; McPeek and Holt 92; Holt and McPeek 1996; Dockery et al. 1998; Kirkland et al. 2006; Abrams 2007; Armsworth and Roughgarden 2008; Amarasekare 2010 Surveys: Johnson and Gaines 1990; Clobert et al. 2001; Levin, Muller-Landau, Nathan and Chave 2003; Bowler and Benton 2005; Holyoak et al. 2005; Amarasekare 2008 Yuan Lou (Ohio State) NC State / 43

6 Unbiased dispersal Evolution game theory Game theory: John von Neumann (28), John Nash (50) Yuan Lou (Ohio State) NC State / 43

7 Unbiased dispersal Evolution game theory Game theory: John von Neumann (28), John Nash (50) Evolutionary game theory: John Maynard Smith and Price (73) Yuan Lou (Ohio State) NC State / 43

8 Unbiased dispersal Evolution game theory Game theory: John von Neumann (28), John Nash (50) Evolutionary game theory: John Maynard Smith and Price (73) Evolutionary stable strategy (ESS): A strategy such that, if all the members of a population adopt it, no mutant strategy can invade Yuan Lou (Ohio State) NC State / 43

9 Unbiased dispersal Evolution game theory Game theory: John von Neumann (28), John Nash (50) Evolutionary game theory: John Maynard Smith and Price (73) Evolutionary stable strategy (ESS): A strategy such that, if all the members of a population adopt it, no mutant strategy can invade Optimal" movement strategy: Dispersal strategies that are evolutionarily stable Yuan Lou (Ohio State) NC State / 43

10 Unbiased dispersal Unbiased dispersal Hastings (TPB, 83); Dockery et al. (JMB, 98) u t = u[m(x) u v] in Ω (0, ), v t = v[m(x) u v] in Ω (0, ), (1) u(x, t), v(x, t): densities at x Ω R N Yuan Lou (Ohio State) NC State / 43

11 Unbiased dispersal Unbiased dispersal Hastings (TPB, 83); Dockery et al. (JMB, 98) u t = u[m(x) u v] in Ω (0, ), v t = v[m(x) u v] in Ω (0, ), (1) u(x, t), v(x, t): densities at x Ω R N m(x): intrinsic growth rate of species Yuan Lou (Ohio State) NC State / 43

12 Unbiased dispersal Unbiased dispersal Hastings (TPB, 83); Dockery et al. (JMB, 98) u t = d 1 u + u[m(x) u v] in Ω (0, ), v t = d 2 v + v[m(x) u v] in Ω (0, ), (1) u(x, t), v(x, t): densities at x Ω R N m(x): intrinsic growth rate of species d 1, d 2 : dispersal rates (strategies) Yuan Lou (Ohio State) NC State / 43

13 Unbiased dispersal Unbiased dispersal Hastings (TPB, 83); Dockery et al. (JMB, 98) u t = d 1 u + u[m(x) u v] v t = d 2 v + v[m(x) u v] in Ω (0, ), in Ω (0, ), (1) u n = v n = 0 on Ω (0, ). u(x, t), v(x, t): densities at x Ω R N m(x): intrinsic growth rate of species d 1, d 2 : dispersal rates (strategies) No-flux boundary condition Yuan Lou (Ohio State) NC State / 43

14 Hasting s approach Unbiased dispersal Suppose that u (resident species) is at equilibrium: d 1 u + u [m(x) u ] = 0 in Ω, u n = 0 on Ω. (2) Question. Can mutant v grow when it is rare? Yuan Lou (Ohio State) NC State / 43

15 Hasting s approach Unbiased dispersal Suppose that u (resident species) is at equilibrium: d 1 u + u [m(x) u ] = 0 in Ω, u n = 0 on Ω. (2) Question. Can mutant v grow when it is rare? Stability of (u, v) = (u, 0): Let Λ(d 1, d 2 ) denote the smallest eigenvalue of d 2 ϕ + (m u )ϕ + λϕ = 0 in Ω, ϕ n = 0 on Ω. Yuan Lou (Ohio State) NC State / 43

16 Unbiased dispersal Evolution of slow dispersal Theorem (Hastings 1983) Suppose that m(x) is non-constant, positive and continuous in Ω. If d 1 < d 2, then (u, 0) is stable; if d 1 > d 2, (u, 0) is unstable. Yuan Lou (Ohio State) NC State / 43

17 Unbiased dispersal Evolution of slow dispersal Theorem (Hastings 1983) Suppose that m(x) is non-constant, positive and continuous in Ω. If d 1 < d 2, then (u, 0) is stable; if d 1 > d 2, (u, 0) is unstable. Λ(d 1, d 1 ) = 0 Yuan Lou (Ohio State) NC State / 43

18 Unbiased dispersal Evolution of slow dispersal Theorem (Hastings 1983) Suppose that m(x) is non-constant, positive and continuous in Ω. If d 1 < d 2, then (u, 0) is stable; if d 1 > d 2, (u, 0) is unstable. Λ(d 1, d 1 ) = 0 Λ(d 1, d 2 ) is increasing in d 2 Yuan Lou (Ohio State) NC State / 43

19 Unbiased dispersal Evolution of slow dispersal Theorem (Hastings 1983) Suppose that m(x) is non-constant, positive and continuous in Ω. If d 1 < d 2, then (u, 0) is stable; if d 1 > d 2, (u, 0) is unstable. Λ(d 1, d 1 ) = 0 Λ(d 1, d 2 ) is increasing in d 2 No dispersal rate is evolutionarily stable: Any mutant with a smaller dispersal rate can invade! Yuan Lou (Ohio State) NC State / 43

20 Unbiased dispersal Ideal free distribution (IFD) Fretwell and Lucas (70) Yuan Lou (Ohio State) NC State / 43

21 Unbiased dispersal Ideal free distribution (IFD) Fretwell and Lucas (70) How should organisms distribute in heterogeneous habitat? Yuan Lou (Ohio State) NC State / 43

22 Unbiased dispersal Ideal free distribution (IFD) Fretwell and Lucas (70) How should organisms distribute in heterogeneous habitat? Assumption 1: Animals are "ideal" in assessment of habitat Yuan Lou (Ohio State) NC State / 43

23 Unbiased dispersal Ideal free distribution (IFD) Fretwell and Lucas (70) How should organisms distribute in heterogeneous habitat? Assumption 1: Animals are "ideal" in assessment of habitat Assumption 2: Animals are capable of moving "freely" Yuan Lou (Ohio State) NC State / 43

24 Unbiased dispersal Ideal free distribution (IFD) Fretwell and Lucas (70) How should organisms distribute in heterogeneous habitat? Assumption 1: Animals are "ideal" in assessment of habitat Assumption 2: Animals are capable of moving "freely" Prediction: Animals aggregate proportionately to the amount of resources Yuan Lou (Ohio State) NC State / 43

25 Unbiased dispersal Ideal free distribution (IFD) Milinski (79) Yuan Lou (Ohio State) NC State / 43

26 Unbiased dispersal Ideal free distribution (IFD) Milinski (79) Yuan Lou (Ohio State) NC State / 43

27 Unbiased dispersal Unbiased dispersal Logistic model u t = d u + u [m(x) u] in Ω (0, ) u n = 0 on Ω (0, ) (3) Yuan Lou (Ohio State) NC State / 43

28 Unbiased dispersal Unbiased dispersal Logistic model u t = d u + u [m(x) u] in Ω (0, ) u n = 0 on Ω (0, ) (3) If u(x, 0) is positive, u(x, t) u (x) as t Yuan Lou (Ohio State) NC State / 43

29 Unbiased dispersal Unbiased dispersal Logistic model u t = d u + u [m(x) u] in Ω (0, ) u n = 0 on Ω (0, ) (3) If u(x, 0) is positive, u(x, t) u (x) as t Does u reach an IFD at equilibrium? That is, m(x) u (x) = constant? Yuan Lou (Ohio State) NC State / 43

30 Unbiased dispersal Heterogeneous environment Logistic model d u + u (m(x) u ) = 0 in Ω, (4) Yuan Lou (Ohio State) NC State / 43

31 Unbiased dispersal Heterogeneous environment Logistic model d u + u (m(x) u ) = 0 in Ω, u n = 0 on Ω (4) No ideal free distribution: m/u constant. Yuan Lou (Ohio State) NC State / 43

32 Unbiased dispersal Heterogeneous environment Logistic model d u + u (m(x) u ) = 0 in Ω, u n = 0 on Ω (4) No ideal free distribution: m/u constant. Integrating (4) in Ω, u (m u ) = 0. Ω Yuan Lou (Ohio State) NC State / 43

33 Unbiased dispersal Heterogeneous environment Logistic model d u + u (m(x) u ) = 0 in Ω, u n = 0 on Ω (4) No ideal free distribution: m/u constant. Integrating (4) in Ω, u (m u ) = 0. If m/u were a constant, then m u. Ω Yuan Lou (Ohio State) NC State / 43

34 Unbiased dispersal Heterogeneous environment Logistic model d u + u (m(x) u ) = 0 in Ω, u n = 0 on Ω (4) No ideal free distribution: m/u constant. Integrating (4) in Ω, u (m u ) = 0. If m/u were a constant, then m u. By (4), Ω m = 0 in Ω, m n = 0 on Ω, which implies that m must be a constant. Contradiction! Yuan Lou (Ohio State) NC State / 43

35 Unbiased dispersal Two competing species Dockery et al. (98) u t = d 1 u + u(m u v) v t = d 2 v + v(m u v) in Ω (0, ), in Ω (0, ), (5) u n = v n = 0 on Ω (0, ). Yuan Lou (Ohio State) NC State / 43

36 Unbiased dispersal Two competing species Dockery et al. (98) u t = d 1 u + u(m u v) v t = d 2 v + v(m u v) in Ω (0, ), in Ω (0, ), (5) u n = v n = 0 on Ω (0, ). Theorem If d 1 < d 2, (u, 0) is globally asymptotically stable. Yuan Lou (Ohio State) NC State / 43

37 Unbiased dispersal Two competing species Dockery et al. (98) u t = d 1 u + u(m u v) v t = d 2 v + v(m u v) in Ω (0, ), in Ω (0, ), (5) u n = v n = 0 on Ω (0, ). Theorem If d 1 < d 2, (u, 0) is globally asymptotically stable. Evolution of slow dispersal: Why? Yuan Lou (Ohio State) NC State / 43

38 Unbiased dispersal Why smaller dispersal rate? Logistic model (6) Yuan Lou (Ohio State) NC State / 43

39 Unbiased dispersal Why smaller dispersal rate? Logistic model d u + u (m(x) u ) = 0 in Ω, u n = 0 on Ω. (6) Yuan Lou (Ohio State) NC State / 43

40 Unbiased dispersal Why smaller dispersal rate? Logistic model It can be shown that d u + u (m(x) u ) = 0 in Ω, u n = 0 on Ω. (6) m(x) lim d 0 u (x) = 1. Yuan Lou (Ohio State) NC State / 43

41 Unbiased dispersal Why smaller dispersal rate? Logistic model It can be shown that d u + u (m(x) u ) = 0 in Ω, u n = 0 on Ω. (6) m(x) lim d 0 u (x) = 1. The smaller d is, the closer m/u to constant; i.e., the distribution of the species is closer to IFD for smaller dispersal rate Yuan Lou (Ohio State) NC State / 43

42 Unbiased dispersal Q: Are there dispersal strategies that can produce ideal free distribution? Yuan Lou (Ohio State) NC State / 43

43 Single species Balanced dispersal Cantrell, Cosner, L (MBE, 10) Yuan Lou (Ohio State) NC State / 43

44 Single species Balanced dispersal Cantrell, Cosner, L (MBE, 10) u t = d 1 [ u u P(x)] + u[m(x) u] in Ω (0, ) Yuan Lou (Ohio State) NC State / 43

45 Single species Balanced dispersal Cantrell, Cosner, L (MBE, 10) u t = d 1 [ u u P(x)] + u[m(x) u] in Ω (0, ) [ u u P(x)] n = 0 on Ω (0, ) (7) Yuan Lou (Ohio State) NC State / 43

46 Single species Balanced dispersal Cantrell, Cosner, L (MBE, 10) u t = d 1 [ u u P(x)] + u[m(x) u] in Ω (0, ) [ u u P(x)] n = 0 on Ω (0, ) (7) P(x) = ln m(x) can produce ideal free distribution Yuan Lou (Ohio State) NC State / 43

47 Balanced dispersal Balanced dispersal If P(x) = ln m(x), then u m is a positive solution of Yuan Lou (Ohio State) NC State / 43

48 Balanced dispersal Balanced dispersal If P(x) = ln m(x), then u m is a positive solution of d 1 [ u u P(x)] + u[m(x) u] = 0 [ u u P(x)] n = 0 on Ω in Ω (8) Yuan Lou (Ohio State) NC State / 43

49 Balanced dispersal Balanced dispersal If P(x) = ln m(x), then u m is a positive solution of d 1 [ u u P(x)] + u[m(x) u] = 0 [ u u P(x)] n = 0 on Ω Ideal free distribution: m u in Ω (8) Yuan Lou (Ohio State) NC State / 43

50 Balanced dispersal Balanced dispersal If P(x) = ln m(x), then u m is a positive solution of d 1 [ u u P(x)] + u[m(x) u] = 0 [ u u P(x)] n = 0 on Ω Ideal free distribution: m u in Ω (8) Balanced dispersal": McPeek and Holt 1992 u u P = m m (ln m) = 0 Yuan Lou (Ohio State) NC State / 43

51 Balanced dispersal Balanced dispersal If P(x) = ln m(x), then u m is a positive solution of d 1 [ u u P(x)] + u[m(x) u] = 0 [ u u P(x)] n = 0 on Ω Ideal free distribution: m u in Ω (8) Balanced dispersal": McPeek and Holt 1992 u u P = m m (ln m) = 0 Is the strategy P = ln m an ESS? Yuan Lou (Ohio State) NC State / 43

52 Two species model Balanced dispersal u t = d 1 [ u u P] + u(m u v) in Ω (0, ) Yuan Lou (Ohio State) NC State / 43

53 Two species model Balanced dispersal u t = d 1 [ u u P] + u(m u v) in Ω (0, ) v t = d 2 [ v v Q] + v(m u v) in Ω (0, ) Yuan Lou (Ohio State) NC State / 43

54 Two species model Balanced dispersal u t = d 1 [ u u P] + u(m u v) in Ω (0, ) v t = d 2 [ v v Q] + v(m u v) in Ω (0, ) (9) [ u u P] n = [ v v Q] n = 0 on Ω (0, ) Yuan Lou (Ohio State) NC State / 43

55 Two species model Balanced dispersal u t = d 1 [ u u P] + u(m u v) in Ω (0, ) v t = d 2 [ v v Q] + v(m u v) in Ω (0, ) (9) [ u u P] n = [ v v Q] n = 0 on Ω (0, ) If P = ln m, (m, 0) is a steady state. Yuan Lou (Ohio State) NC State / 43

56 Two species model Balanced dispersal u t = d 1 [ u u P] + u(m u v) in Ω (0, ) v t = d 2 [ v v Q] + v(m u v) in Ω (0, ) (9) [ u u P] n = [ v v Q] n = 0 on Ω (0, ) If P = ln m, (m, 0) is a steady state. Is (m, 0) asymptotically stable? ( Is P = ln m an ESS?) Yuan Lou (Ohio State) NC State / 43

57 Stability of (m, 0) Balanced dispersal Original system: u t = d 1 [ u u ln m] + u(m u v), v t = d 2 [ v v Q] + v(m u v). (10) Yuan Lou (Ohio State) NC State / 43

58 Stability of (m, 0) Balanced dispersal Original system: u t = d 1 [ u u ln m] + u(m u v), v t = d 2 [ v v Q] + v(m u v). (10) Perturbation of (m(x), 0): (u, v) = (m, 0) + (ɛϕ(x)e λt, ɛψ(x)e λt ) Yuan Lou (Ohio State) NC State / 43

59 Stability of (m, 0) Balanced dispersal Original system: u t = d 1 [ u u ln m] + u(m u v), v t = d 2 [ v v Q] + v(m u v). (10) Perturbation of (m(x), 0): (u, v) = (m, 0) + (ɛϕ(x)e λt, ɛψ(x)e λt ) Equations for (ϕ, ψ, λ): d 1 [ ϕ ϕ ln m] mϕ mψ = λϕ, d 2 [ ψ ψ Q] = λψ. (11) Yuan Lou (Ohio State) NC State / 43

60 Stability of (m, 0) Balanced dispersal Yuan Lou (Ohio State) NC State / 43

61 Stability of (m, 0) Balanced dispersal Eigenvalue problem for the stability of (m, 0): d 2 [ ψ ψ Q] = λψ in Ω, [ ψ ψ Q] = 0 on Ω. Yuan Lou (Ohio State) NC State / 43

62 Stability of (m, 0) Balanced dispersal Eigenvalue problem for the stability of (m, 0): d 2 [ ψ ψ Q] = λψ in Ω, [ ψ ψ Q] = 0 on Ω. (λ, ψ) = (0, e Q ) is a solution Yuan Lou (Ohio State) NC State / 43

63 Stability of (m, 0) Balanced dispersal Eigenvalue problem for the stability of (m, 0): d 2 [ ψ ψ Q] = λψ in Ω, [ ψ ψ Q] = 0 on Ω. (λ, ψ) = (0, e Q ) is a solution Bad news: Zero is the smallest eigenvalue; i.e., (m, 0) is neutrally stable Yuan Lou (Ohio State) NC State / 43

64 Balanced dispersal Evolutionary stable strategy Cantrell et. al (10); Averill, Munther, L (JBD, 2012) Theorem Suppose that m C 2 ( Ω), is non-constant and positive in Ω. If P = ln m and Q ln m is non-constant, then (m, 0) is globally stable. P = lnm is an ESS: It can resist the invasion of any other strategy Yuan Lou (Ohio State) NC State / 43

65 Balanced dispersal Evolutionary stable strategy Cantrell et. al (10); Averill, Munther, L (JBD, 2012) Theorem Suppose that m C 2 ( Ω), is non-constant and positive in Ω. If P = ln m and Q ln m is non-constant, then (m, 0) is globally stable. P = lnm is an ESS: It can resist the invasion of any other strategy It can displace any other strategy Yuan Lou (Ohio State) NC State / 43

66 Proof Balanced dispersal Yuan Lou (Ohio State) NC State / 43

67 Proof Balanced dispersal Define E(t) = Ω [u(x, t) + v(x, t) m(x) ln u(x, t)] dx. Then de/dt 0 for all t 0. Yuan Lou (Ohio State) NC State / 43

68 Proof Balanced dispersal Define E(t) = Ω [u(x, t) + v(x, t) m(x) ln u(x, t)] dx. Then de/dt 0 for all t 0. Three or more competing species: Gejji et al. (BMB 2012); Munther and L. (DCDS-A 2012) Yuan Lou (Ohio State) NC State / 43

69 Balanced dispersal Other dispersal strategies which can produce ideal free distribution: (Mark Lewis) ( u ) u t = d + u[m(x) u] (12) m Yuan Lou (Ohio State) NC State / 43

70 Balanced dispersal Other dispersal strategies which can produce ideal free distribution: (Mark Lewis) ( u ) u t = d + u[m(x) u] (12) m (Dan Ryan) [ ( u )] u t = d mf (m, m) + u[m(x) u], (13) m where f (m(x 1 ), m(x 2 )) is the probability moving from x 1 to x 2 which satisfies D 2 f (m, m) D 1 f (m, m) = f (m, m) m. Yuan Lou (Ohio State) NC State / 43

71 Single species Balanced dispersal Cosner, Davilla and Martinez (JBD, 11) u t = k(x, y)u(y, t) dy u(x, t) k(y, x) dy + u[m(x) u] (14) Ω Ω Yuan Lou (Ohio State) NC State / 43

72 Single species Balanced dispersal Cosner, Davilla and Martinez (JBD, 11) u t = k(x, y)u(y, t) dy u(x, t) k(y, x) dy + u[m(x) u] (14) Ω Ω Definition: k(x, y) is an ideal free dispersal strategy if k(x, y)m(y) dy = m(x) k(y, x) dy, x Ω. (15) Ω Ω Yuan Lou (Ohio State) NC State / 43

73 Single species Balanced dispersal Cosner, Davilla and Martinez (JBD, 11) u t = k(x, y)u(y, t) dy u(x, t) k(y, x) dy + u[m(x) u] (14) Ω Ω Definition: k(x, y) is an ideal free dispersal strategy if k(x, y)m(y) dy = m(x) k(y, x) dy, x Ω. (15) Ω Ω Example: k(x, y) = m τ (x)m τ 1 (y). Yuan Lou (Ohio State) NC State / 43

74 Single species Balanced dispersal Cosner, Davilla and Martinez (JBD, 11) u t = k(x, y)u(y, t) dy u(x, t) k(y, x) dy + u[m(x) u] (14) Ω Ω Definition: k(x, y) is an ideal free dispersal strategy if k(x, y)m(y) dy = m(x) k(y, x) dy, x Ω. (15) Ω Ω Example: k(x, y) = m τ (x)m τ 1 (y). m(x) is an equilibrium of (14) k(x, y) satisfies (15). Yuan Lou (Ohio State) NC State / 43

75 Two species model Balanced dispersal Cantrell, Cosner, L and Ryan (Canadian Appl. Math. Quart., in press) u t = v t = Ω Ω k(x, y)u(y, t) dy u(x, t) Ω k (x, y)v(y, t) dy v(x, t) Ω k(y, x) dy + u[m(x) u v], k (y, x) dy + v[m(x) u v]. (16) Yuan Lou (Ohio State) NC State / 43

76 Two species model Balanced dispersal Cantrell, Cosner, L and Ryan (Canadian Appl. Math. Quart., in press) u t = v t = Ω Ω k(x, y)u(y, t) dy u(x, t) Ω k (x, y)v(y, t) dy v(x, t) Ω k(y, x) dy + u[m(x) u v], k (y, x) dy + v[m(x) u v]. (16) Theorem Suppose that both k and k are continuous and positive in Ω Ω, k is an ideal free dispersal strategy and k is not an ideal dispersal strategy. Then, (m(x), 0) of (16) is globally stable in C( Ω) C( Ω) for all positive initial data. Yuan Lou (Ohio State) NC State / 43

77 A key ingredient Balanced dispersal Let h : Ω Ω [0, ) be a continuous function. Then the following two statements are equivalent: Yuan Lou (Ohio State) NC State / 43

78 A key ingredient Balanced dispersal Let h : Ω Ω [0, ) be a continuous function. Then the following two statements are equivalent: i h(x, y) dy = Ω Ω h(y, x) dy for all x Ω. Yuan Lou (Ohio State) NC State / 43

79 A key ingredient Balanced dispersal Let h : Ω Ω [0, ) be a continuous function. Then the following two statements are equivalent: i h(x, y) dy = Ω Ω h(y, x) dy for all x Ω. ii Ω Ω f (x) f (y) h(x, y) dx dy 0 for any f C(Ω), f > 0 on Ω. f (y) Yuan Lou (Ohio State) NC State / 43

80 Summary Balanced dispersal Yuan Lou (Ohio State) NC State / 43

81 Summary Balanced dispersal Balanced dispersal strategies are generally evolutionarily stable Yuan Lou (Ohio State) NC State / 43

82 Summary Balanced dispersal Balanced dispersal strategies are generally evolutionarily stable What happens if dispersal strategies are unbalanced? Yuan Lou (Ohio State) NC State / 43

83 Single species Biased dispersal Yuan Lou (Ohio State) NC State / 43

84 Single species Biased dispersal Biased dispersal: Organisms can sense and respond to local environmental cues Yuan Lou (Ohio State) NC State / 43

85 Single species Biased dispersal Biased dispersal: Organisms can sense and respond to local environmental cues Belgacem and Cosner (Canadian Appl. Math Quart. 1995) Yuan Lou (Ohio State) NC State / 43

86 Single species Biased dispersal Biased dispersal: Organisms can sense and respond to local environmental cues Belgacem and Cosner (Canadian Appl. Math Quart. 1995) u t = d 1 [ u αu m] + u(m u) in Ω (0, ), Yuan Lou (Ohio State) NC State / 43

87 Single species Biased dispersal Biased dispersal: Organisms can sense and respond to local environmental cues Belgacem and Cosner (Canadian Appl. Math Quart. 1995) u t = d 1 [ u αu m] + u(m u) in Ω (0, ), [ u αu m] n = 0 on Ω (0, ) (17) Yuan Lou (Ohio State) NC State / 43

88 Biased vs unbiased Biased dispersal Yuan Lou (Ohio State) NC State / 43

89 Biased vs unbiased Biased dispersal Cantrell, Cosner and L. (Math. Bios., 06) Yuan Lou (Ohio State) NC State / 43

90 Biased vs unbiased Biased dispersal Cantrell, Cosner and L. (Math. Bios., 06) Yuan Lou (Ohio State) NC State / 43

91 Biased vs unbiased Biased dispersal Cantrell, Cosner and L. (Math. Bios., 06) u t = d 1 [ u αu m] + u(m u v) in Ω (0, ), Yuan Lou (Ohio State) NC State / 43

92 Biased vs unbiased Biased dispersal Cantrell, Cosner and L. (Math. Bios., 06) u t = d 1 [ u αu m] + u(m u v) in Ω (0, ), v t = d 2 v + v(m u v) in Ω (0, ), Yuan Lou (Ohio State) NC State / 43

93 Biased vs unbiased Biased dispersal Cantrell, Cosner and L. (Math. Bios., 06) u t = d 1 [ u αu m] + u(m u v) in Ω (0, ), v t = d 2 v + v(m u v) in Ω (0, ), [ u αu m] n = v n = 0 on Ω (0, ) (18) Yuan Lou (Ohio State) NC State / 43

94 Weak advection Biased dispersal Cantrell, Cosner and L. (Proc. Roy Soc. Edin, 07) Theorem Suppose that m C 2 ( Ω), positive, non-constant. Yuan Lou (Ohio State) NC State / 43

95 Weak advection Biased dispersal Cantrell, Cosner and L. (Proc. Roy Soc. Edin, 07) Theorem Suppose that m C 2 ( Ω), positive, non-constant. If d 1 = d 2, α > 0 small and Ω is convex, (u, 0) is globally stable Yuan Lou (Ohio State) NC State / 43

96 Weak advection Biased dispersal Cantrell, Cosner and L. (Proc. Roy Soc. Edin, 07) Theorem Suppose that m C 2 ( Ω), positive, non-constant. If d 1 = d 2, α > 0 small and Ω is convex, (u, 0) is globally stable For some non-convex Ω and m(x), (0, v ) is globally stable Yuan Lou (Ohio State) NC State / 43

97 Strong advection Biased dispersal Cantrell et al. (07); Chen, Hambrock, L (JMB, 08) Yuan Lou (Ohio State) NC State / 43

98 Strong advection Biased dispersal Cantrell et al. (07); Chen, Hambrock, L (JMB, 08) Theorem Suppose that m C 2 ( Ω), positive, non-constant. Yuan Lou (Ohio State) NC State / 43

99 Strong advection Biased dispersal Cantrell et al. (07); Chen, Hambrock, L (JMB, 08) Theorem Suppose that m C 2 ( Ω), positive, non-constant. For any d 1 and d 2, if α is large, both (u, 0) and (0, v ) are unstable, and system (18) has a stable positive steady state. Yuan Lou (Ohio State) NC State / 43

100 Strong advection Biased dispersal Cantrell et al. (07); Chen, Hambrock, L (JMB, 08) Theorem Suppose that m C 2 ( Ω), positive, non-constant. For any d 1 and d 2, if α is large, both (u, 0) and (0, v ) are unstable, and system (18) has a stable positive steady state. Strong advection can induce coexistence of competing species Yuan Lou (Ohio State) NC State / 43

101 Aggregation Biased dispersal Theorem Let (u, v) be a positive steady state of system (18). As α, v(x) v and { } u(x) = e α[m(x 0) m(x)] 2 N 2 [m(x0 ) v (x 0 )] + o(1), where x 0 is a local maximum of m such that m(x 0 ) v (x 0 ) > 0. Yuan Lou (Ohio State) NC State / 43

102 Aggregation Biased dispersal Theorem Let (u, v) be a positive steady state of system (18). As α, v(x) v and { } u(x) = e α[m(x 0) m(x)] 2 N 2 [m(x0 ) v (x 0 )] + o(1), where x 0 is a local maximum of m such that m(x 0 ) v (x 0 ) > 0. Chen and L (Indiana Univ. Math J, 08): m has a unique local maximum Yuan Lou (Ohio State) NC State / 43

103 Aggregation Biased dispersal Theorem Let (u, v) be a positive steady state of system (18). As α, v(x) v and { } u(x) = e α[m(x 0) m(x)] 2 N 2 [m(x0 ) v (x 0 )] + o(1), where x 0 is a local maximum of m such that m(x 0 ) v (x 0 ) > 0. Chen and L (Indiana Univ. Math J, 08): m has a unique local maximum Lam and Ni (DCDS-A, 10): m finite many local maxima, N = 1 Yuan Lou (Ohio State) NC State / 43

104 Aggregation Biased dispersal Theorem Let (u, v) be a positive steady state of system (18). As α, v(x) v and { } u(x) = e α[m(x 0) m(x)] 2 N 2 [m(x0 ) v (x 0 )] + o(1), where x 0 is a local maximum of m such that m(x 0 ) v (x 0 ) > 0. Chen and L (Indiana Univ. Math J, 08): m has a unique local maximum Lam and Ni (DCDS-A, 10): m finite many local maxima, N = 1 Lam (SIMA, 12): m finite many local maxima, N 1 Yuan Lou (Ohio State) NC State / 43

105 Biased dispersal Consider u t = d [ u αu m] + u(m u v) in Ω (0, ), v t = d [ v βv m] + v(m u v) in Ω (0, ), (19) [ u αu m] n = [ v βv m] n = 0 on Ω Question. Can we find some advection rate which is evolutionarily stable? Yuan Lou (Ohio State) NC State / 43

106 Biased dispersal Hasting s approach revisited Suppose that species u is at equilibrium: d [ u αu m] + u [m(x) u ] = 0 in Ω, [ u αu m] n = 0 on Ω. (20) Question. Can species v grow when it is rare? Yuan Lou (Ohio State) NC State / 43

107 Biased dispersal Hasting s approach revisited Suppose that species u is at equilibrium: d [ u αu m] + u [m(x) u ] = 0 in Ω, [ u αu m] n = 0 on Ω. (20) Question. Can species v grow when it is rare? Stability of (u, v) = (u, 0): Let Λ(α, β) denote the smallest eigenvalue of d [ ϕ βϕ m] + (m u )ϕ + λϕ = 0 in Ω, [ ϕ βϕ m] n = 0 on Ω. Yuan Lou (Ohio State) NC State / 43

108 Adaptive Dynamics Biased dispersal Question: Is there an ESS? That is, there exists some α > 0 such that Λ(α, β) > 0, β α Yuan Lou (Ohio State) NC State / 43

109 Adaptive Dynamics Biased dispersal Question: Is there an ESS? That is, there exists some α > 0 such that Λ(α, β) > 0, β α Yuan Lou (Ohio State) NC State / 43

110 Adaptive Dynamics Biased dispersal Question: Is there an ESS? That is, there exists some α > 0 such that Λ(α, β) > 0, β α Step 1. Find α such that Λ β (α, α ) = 0. Such α is called evolutionarily singular strategy. Yuan Lou (Ohio State) NC State / 43

111 Adaptive Dynamics Biased dispersal Question: Is there an ESS? That is, there exists some α > 0 such that Λ(α, β) > 0, β α Step 1. Find α such that Λ β (α, α ) = 0. Such α is called evolutionarily singular strategy. Step 2. If α is an evolutionarily singular strategy, determine the sign of 2 Λ β 2 (α, α ) Yuan Lou (Ohio State) NC State / 43

112 K.-Y. Lam and L. (2012) Biased dispersal Theorem Suppose that m > 0, C 2 ( Ω), 1 < max Ω m min Ω m Given any γ > 0, if d is small, there exists exactly exactly one evolutionarily singular strategy, denoted as α, in (0, γ]. As d 0, α η, where η is the unique positive root of e ηm (1 ηm)m m 2 = 0. Ω For some functions m satisfying max Ω m min Ω m > , there are at least 3 evolutionarily singular strategies. Yuan Lou (Ohio State) NC State / 43

113 Biased dispersal Theorem Suppose that Ω is convex and ln(m) L α 0 diam(ω), where α , then for small d, α = α, β α and β α, (u, 0) is asymptotically stable. max Ω m min Ω m eα < For some function m satisfying max Ω m min Ω m > , there exists some evolutionarily singular strategy which is not an ESS. Yuan Lou (Ohio State) NC State / 43

114 Biased dispersal One ingredient of the proof is the following celebrated theorem of Payne and Weinberger: Theorem Suppose that Ω is a convex domain in R N. Let µ 2 denote the second eigenvalue of the Laplacian with Neumann boundary condition. Then ( ) π 2 µ 2. diam(ω) Yuan Lou (Ohio State) NC State / 43

115 Single species Fitness-dependent dispersal Yuan Lou (Ohio State) NC State / 43

116 Single species Fitness-dependent dispersal Dispersal up the gradient of fitness: Armsworth and Roughgarden 2005, 2008; Abrams 2007; Amarasekare 2010 Yuan Lou (Ohio State) NC State / 43

117 Single species Fitness-dependent dispersal Dispersal up the gradient of fitness: Armsworth and Roughgarden 2005, 2008; Abrams 2007; Amarasekare 2010 Cosner (TPB 2005); Cantrell, Cosner, L. (JDE 2008); Cosner and Winkler (2013) Yuan Lou (Ohio State) NC State / 43

118 Single species Fitness-dependent dispersal Dispersal up the gradient of fitness: Armsworth and Roughgarden 2005, 2008; Abrams 2007; Amarasekare 2010 Cosner (TPB 2005); Cantrell, Cosner, L. (JDE 2008); Cosner and Winkler (2013) u t = d [ u αu (m u)] + u(m u) in Ω (0, ), Yuan Lou (Ohio State) NC State / 43

119 Single species Fitness-dependent dispersal Dispersal up the gradient of fitness: Armsworth and Roughgarden 2005, 2008; Abrams 2007; Amarasekare 2010 Cosner (TPB 2005); Cantrell, Cosner, L. (JDE 2008); Cosner and Winkler (2013) u t = d [ u αu (m u)] + u(m u) in Ω (0, ), [ u αu (m u)] n = 0 on Ω (0, ) Yuan Lou (Ohio State) NC State / 43

120 Fitness-dependent dispersal Fitness-dependent vs unbiased dispersal Yuan Lou (Ohio State) NC State / 43

121 Fitness-dependent dispersal Fitness-dependent vs unbiased dispersal Cantrell, Cosner, L. and Xie (JDE 2013); L. Tao and Winkler (2013) Yuan Lou (Ohio State) NC State / 43

122 Fitness-dependent dispersal Fitness-dependent vs unbiased dispersal Cantrell, Cosner, L. and Xie (JDE 2013); L. Tao and Winkler (2013) Yuan Lou (Ohio State) NC State / 43

123 Fitness-dependent dispersal Fitness-dependent vs unbiased dispersal Cantrell, Cosner, L. and Xie (JDE 2013); L. Tao and Winkler (2013) u t = d 1 [ u αu (m u v)] + u(m u v), Yuan Lou (Ohio State) NC State / 43

124 Fitness-dependent dispersal Fitness-dependent vs unbiased dispersal Cantrell, Cosner, L. and Xie (JDE 2013); L. Tao and Winkler (2013) u t = d 1 [ u αu (m u v)] + u(m u v), v t = d 2 v + v(m u v) in Ω (0, ), Yuan Lou (Ohio State) NC State / 43

125 Fitness-dependent dispersal Fitness-dependent vs unbiased dispersal Cantrell, Cosner, L. and Xie (JDE 2013); L. Tao and Winkler (2013) u t = d 1 [ u αu (m u v)] + u(m u v), v t = d 2 v + v(m u v) in Ω (0, ), [ u αu (m u v)] n = v n = 0 on Ω (0, ) Yuan Lou (Ohio State) NC State / 43

126 Fitness-dependent dispersal Recent developments Convergent stable strategy, evolution of two traits: Gejji et al, BMB, 2012 Yuan Lou (Ohio State) NC State / 43

127 Fitness-dependent dispersal Recent developments Convergent stable strategy, evolution of two traits: Gejji et al, BMB, 2012 Directed movement in periodic environment: Kawasaki et al., BMB, 2012 Yuan Lou (Ohio State) NC State / 43

128 Fitness-dependent dispersal Recent developments Convergent stable strategy, evolution of two traits: Gejji et al, BMB, 2012 Directed movement in periodic environment: Kawasaki et al., BMB, 2012 Multi-trophic level models: X.-F. Wang and Y.-P. Wu 2002; D. DeAngelis et al. Am. Nat, 2011; Wu and L, SIAP 2011 Yuan Lou (Ohio State) NC State / 43

129 Fitness-dependent dispersal Recent developments Convergent stable strategy, evolution of two traits: Gejji et al, BMB, 2012 Directed movement in periodic environment: Kawasaki et al., BMB, 2012 Multi-trophic level models: X.-F. Wang and Y.-P. Wu 2002; D. DeAngelis et al. Am. Nat, 2011; Wu and L, SIAP 2011 Dispersal in random environments: Evans et al. JMB 2012; S. Schreiber, Am. Nat, in press Yuan Lou (Ohio State) NC State / 43

130 Fitness-dependent dispersal Acknowledgment Collaborators: Steve Cantrell, Chris Cosner (University of Miami) Isabel Averill, Richard Hambrock Xinfu Chen (University of Pittsburgh) King-Yeung Lam (MBI) Dan Munther (York University) Dan Ryan (NIMBioS) Yuan Lou (Ohio State) NC State / 43

131 Fitness-dependent dispersal Acknowledgment Collaborators: Steve Cantrell, Chris Cosner (University of Miami) Isabel Averill, Richard Hambrock Xinfu Chen (University of Pittsburgh) King-Yeung Lam (MBI) Dan Munther (York University) Dan Ryan (NIMBioS) Support: NSF, Mathematical Biosciences Institute Yuan Lou (Ohio State) NC State / 43

132 Fitness-dependent dispersal MBI Emphasis Year on Cancer and Its Environment: Ecology and Evolution of Cancer Metastasis and Angiogenesis Cancer and the Immune System Tumor Heterogeneity and the Microenvironment Treatment, Clinical Trials, Resistance Targeting Cancer Cell Proliferation and Metabolism Networks Stem Cells, Development, and Cancer Yuan Lou (Ohio State) NC State / 43

133 Fitness-dependent dispersal MBI Emphasis Year on Cancer and Its Environment: Ecology and Evolution of Cancer Metastasis and Angiogenesis Cancer and the Immune System Tumor Heterogeneity and the Microenvironment Treatment, Clinical Trials, Resistance Targeting Cancer Cell Proliferation and Metabolism Networks Stem Cells, Development, and Cancer Thank you! Yuan Lou (Ohio State) NC State / 43