A Comparison of Two Predator-Prey Models with Holling s Type I Functional Response ** Joint work with Mark Kot at the University of Washington ** Mathematical Biosciences 1 (8) 161-179 Presented by Gunog Seo York University / Ryerson University
Model Formulation intrinsic rate of growth of the prey dn dt = rn 1 N K Functional Response Φ(N)P dp dt = G(N,P)P Numerical Response Carrying capacity N(T ) = Prey population P (T ) = Predator population
Model Formulation dn dt dp dt = rn 1 N K = G(N,P)P Φ(N)P 3 Functional Responses: identified by C. S. Holling (1959, 1965, 1966) c c c!(n) c/!(n) c/!(n) c/ Type I Type II Type III a a N a a N a a N Φ(N) = c a N if N<a c if N a. Φ(N) = cn a + N Φ(N) = cn a + N
Model Formulation dn dt dp dt = rn 1 N K = G(N,P)P Φ(N)P 4 Numerical Responses Laissez-faire Leslie G(N,P) = b c Φ(N) m G(N,P) =s 1 P hn
Outline Φ I (N) = { c a N, N < a, c, N a. Laissez-faire Model with a type I functional response. Leslie-type Model with a type I functional response. Nondimensionalization. Stability analyses of equilibria: performing a linearized stability analysis, constructing a Lyapunov function. Numerical studies. if time permits Laissez-faire and Leslie-type models with arctan functional responses. Laissez-faire Model dn dt = rn dp dt = P dp dt = sp 1 N Φ I (N)P K b c Φ I(N) m Leslie-type Model dn dt = rn 1 N Φ I (N)P K 1 P hn Discussion
6 Models with Type I Functional Responses D. M. Dubois and P. L. Closset. Patchiness in primary and secondary production in the Southern Bight: a mathematical theory. In G. Persoone and E. Jaspers, editors, Proceedings of the 1th European Symposium on Marine Biology, pp. 11!9, Universa Press, Ostend, 1976. Y. Ren and L. Han. The predator prey model with two limit cycles. Acta Math. Appl. Sinica (English Ser.), 5: 3!3, 1989. J. B. Collings. The effects of the functional response on the bifurcation behavior of a mite predator!prey interaction model. J. Math. Biol., 36: 149!168, 1997. G. Dai and M. Tang. Coexistence region and global dynamics of a harvested predator!prey system. SIAM J. Appl. Math., 58: 193!1, 1998. X. Y. Li and W. D. Wang. Qualitative analysis of predator!prey system with Holling type I functional response. J. South China Normal Univ. (Natur. Sci. Ed.), 9: 71!717, 4. B. Liu, Y. Zhang, and L. Chen. Dynamics complexities of a Holling I predator!prey model concerning periodic biological and chemical control. Chaos Solitons Fractals, : 13!134, 4. Y. Zhang, Z. Xu, and B. Liu. Dynamic analysis of a Holling I predator!prey system with mutual interference concerning pest control. J. Biol. Syst., 13:45!58, 5.
dn dt = rn dp dt = P 1 N Φ I (N)P K b c Φ I(N) m Laissez-faire Model with a type I functional response
Laissez-faire Model with a type I functional response Nondimensionalization & Equilibria 8 dn dt = rn 1 N Φ I (N)P K dp b dt = P c Φ I(N) m c where Φ I (N) = a N c if N<a if N a x = N a, y = c ra P, t = rt α = b r, β = m b, γ = K a dx dt = x 1 x φ I (x)y γ dy dt = αy(φ I(x) β) where φ I (x) = x/, x < 1, x Assuming α >, < β < 1, and γ > Equilibria E =(, ), E 1 =(γ, ), E =(β,g(β)) where g(x) = x φ I (x) 1 x γ
Laissez-faire Model with a type I functional response Linearized Stability Analysis 9 Assuming α >, < β < 1, and γ > Equilibria E =(, ), E 1 =(γ, ), saddle point saddle point if φ I (γ) > β stable node if φ I (γ) < β E =(β,g(β)) where g(x) = x φ I (x) 1 x γ J = φ I (x)(g(x) y)+φ I (x)g (x) φ I (x) αyφ I (x) α (φ I(x) β)
Laissez-faire Model with a type I functional response Linearized Stability Analysis 1 Assuming α >, < β < 1, and γ > E =(, ), saddle point Equilibria E 1 =(γ, ), saddle point if φ I (γ) > β stable node if φ I (γ) < β E =(β,g(β)) where g(x) = x φ I (x) 1 x γ By Routh-Hurwitz criterion, J E = the coexistence equilibrium is asymptotically stable. βg (β) β αg(β)φ I (β) Characteristic Equation λ βg (β)λ + αβg(β)φ I(β) =
Laissez-faire Model with a type I functional response Global Stability Analysis of E 11 Using Harrison s Gedankenexperiment (1979), construct Lyapunov function where g(x) = V (x, y) = x 1 x φ I (x) γ x α(φ I (ξ) β) dξ + α x φ I (ξ) and x =β V (x, y) =V 1 (x)+v (x) where V 1 (x) =α V = dv dt g(x ) y g(x ) ξ αξ dξ (x x ) x ln x x x<, ( x ) x ln x +(1 β)(x ), x V (x) =g(x )ln g(x ) + y g(x ) y = α (φ I (x) β)(g(x) g(x )) is continuous at x = V (x, y) is continuous at x = and is zero at the coexistence equilibrium For all positive x and y, V (x, y) is positive, except at coexistence equilibrium E. V < in a neighborhood of E if g(x) >g(x ) for x<x AND g(x) <g(x ) for x>x.
Laissez-faire Model with a type I functional response Global Stability Analysis of E 1 (a) y The coexistence equilibrium is globally asymptotically stable. A subset of the basin of attraction x x γ D = {(x, y) V (x, y) <u} (b) where u = min {V (,g(x )), V(x M,g(x )), V(x, ), V(x, + )} ULC SLC with x M = γ γ 8(γ β) when γ > 4+4 1 β. y D x x γ
Laissez-faire Model with a type I functional response Numerical Studies 13 Poincaré or first return map (!, g(!)) x P(x ) = x 1 Poincar é section ( ) : y = 1 β γ for x > β
Laissez-faire Model with a type I functional response Numerical Studies α =, γ =8 14 Poincaré or first return map 8 7 8 7 8 7 P (x) 6 5 4 3 1 β =.5 4 6 8 x P (x) 6 5 4 3 β =.65 4 6 8 x P (x) 6 5 4 3 β =.85 4 6 8 x
Norm " 4.5 3.5.5 Laissez-faire Model with a type I functional response 5 4 3 Numerical Studies CFB 4 6 8 1 1 9 8.8 8.6 8.4 8. 8 7.8 α = 7 α = α =.5.1..3.4.5.6.7 # above each curve. (a)! Norm " 5 4 9 8.8 3 CFB 8.6 8.4.5.6.7.8.9 8.! 8 7.8 1 α = 7 α = α =.5 (a) 47 Norm 6 5 4 9 CFB (b) 3 Bifurcation 8.6 Diagrams (using 8.4 XPPAUT) Norm 8..5.6.7.8.9 8 β =.7, γ =8 α =, γ =8 7.8 α =, β =.7 (a).1..3 (c).4.5.6.7 6 CFB! 1 Two limit cycles occur # 8 61 4 8 CFB! 6 6 8 1 1 14 16 4 Figure.6: Bifurcation diagrams for laissez-faire model (.6). The L norms of s cycles (filled circles), unstable.4 limit.5cycles.6 (open circles),.7 and.8 stable.9 coexistence! (thick solid line) are computed for different values! Two limit cycles of (a) occur α (β =.7, γ = 8), (b) γ = 8), and (c) γ (α =, β =.7). Two global cyclic-fold bifurcations occur as to the right of each curve. (a) α and one cyclic-fold bifurcation appears as I increase (b) β or (c) γ..1..3.4.5.6.7 8 6 4 Two limit cycles occur to the right of each curve. γ = 8.5 (b) γ = 8 γ = 7.7.4.5.6.7.8.9! " # # 8.8 1 8 6 4 α = 7 α = α =.5 γ = 8.5 β =.7 β =.7 (a) (b) " γ = 8 (c) 47 γ = 7.7 β =.68 7.8 7.85 7.9 7.95 8 8.5 8.1 8.15 " 15 1 (b) 1 (c) Figure.7: Two-parameter bifurcation diagrams for laissez-faire model (.6) indicate cyclic-fold bifurcations. Two limit cycles occur above each curve in (
dn dt = rn dp dt = sp 1 N Φ I (N)P K 1 P hn Leslie-type Model with a type I functional response
Leslie-type Model with a type I functional response Nondimensionalization & Equilibria 17 dn dt = rn 1 N Φ I (N)P K dp dt = sp 1 P hn c where Φ I (N) = a N c if N<a if N a x = N a, y = c ra P, t = rt A = s r, B = ch r, γ = K a where φ I (x) = 1 x φ I (x)y γ dx dt = x dy dt = Ay 1 y Bx x/, x < 1, x Assuming A>, B >, and γ > focus on the dynamics in <x(t) γ, where a unique coexistence equilibrium exists. Equilibria E 1 =(γ, ) E =(ˆx, ŷ ) where ˆx = γ Bγ+, γ(1 B) <, γ(1 B), γ(1 B) ŷ = Bˆx = g(ˆx ) with g(x) = x 1 x φ I (x) γ
Leslie-type Model with a type I functional response Stability Analysis where φ I (x) = 1 x φ I (x)y γ dx dt = x dy dt = Ay 1 y ˆx = Bx x/, x < 1, x { γ Bγ+, γ(1 B) <, γ(1 B), γ(1 B). Equilibria! φ J = I (x)(g(x) y)+φ I (x)g (x) E 1 =(γ, ) E =(ˆx, ŷ ) Ê =(ˆx, ŷ ) A B φ I (x) y x A 1 y Bx saddle point The left (J ) and right (J + ) Jacobians evaluated at Ê where < ˆx < or< ˆx < γ, ( ) φi (ˆx J ± = )g ±(ˆx ) φ I (ˆx ) (ˆx,ŷ ) AB A with ŷ = g(ˆx )= ˆx (1 ˆx /γ) φ I (ˆx ) and { g (ˆx ) = /γ, < ˆx <, g +(ˆx ) = 1 ˆx /γ, < ˆx < γ. 18 λ + λ(a φ I (ˆx )g ±(ˆx )) + Aφ I (ˆx ) ( B g ±(ˆx ) ) = > Asymptotically stable where < ˆx < or γ/ ˆx < γ For < ˆx < γ/, Stable if A>g (ˆx ) Unstable if A<g (ˆx )
Leslie-type Model with a type I functional response Stability Analysis 19!"#$%&'((!)(*+#,!"#-./"%&'((!)(*+#, The generalized Jacobian of Clarke (1998) y Ê J Σ = γ/ x ( J 11 Ê 1 γ Σ 1 AB A ) J Σ = {(1 q)j + qj +, q 1} : a convex combination of the left and right Jacobian, ( ) φi (ˆx J ± = )g ±(ˆx ) φ I (ˆx ) (ˆx,ŷ ) AB A with J 11 Σ = qb +(B 1) where <B<1 and q 1. Characteristic Equation: λ J 11 Σ A λ + A(B J 11 Σ )= λ 1, = (J 11 Σ A) ± i 4AB (J 11 Σ + A) A discontinuous Hopf bifurcation (Leine and Nijmeijer; 4) is expected when the eigenvalues are imaginary(jσ 11 = A) i.e., when q = A B +1 where A B 1 < 1. B
(a). Leslie-type Model with a type I functional response The generalized Jacobian of Clarke!. (b).5 (c) A =.1 q =( ), q =1( ).. The generalized Jacobian of Clarke at ˆx =!.5.5 (d) λ 1 λ B =.43!..5!.!.5.5 (e).!.5.5 (f) (f) B =.85!...5!.5.5!.!.5.5 lues of generalized Jacobian (.38) evaluated at x = in Leslie-.1: (a) B Figure =.8,.8: (b) Paths B = of.3, eigenvalues (c) B = of.43, generalized (d) B = Jacobian.5, (e)(.38) evaluated at x = in Leslie-
Leslie-type Model with a type I functional response Numerical Studies 1 (a) (b).5 D L.5 D L A =1,B =, γ =6 A =1,B =.18, γ =9 1.5 y y 1.5 1 1.5.5 y 4 6 4 3.5 3.5 1.5 1.5 D L x SLC ULC (c) 4 6 8 A =.5,B =1.4, γ = 1.5 A =.1,B =.35, γ = 1 y 3.5 3.5 1.5 1.5 x (d) D L 4 6 8 1 x 4 6 8 1 x
Leslie-type Model with a type I functional response Numerical Studies: Two-parameter bifurcation diagrams A =.1! 18 16 14 1 1 8 6 L L 1 L L.5 1 1.5.5 3 3.5 B H + DHB CFB A.6.5.4.3..1 L L 1 L (b).4.6.8 1 1. 1.4 1.6 B L γ = 1 B =.9 A.8.6.4. L L 5 1 15 5 3! L 1 The parameter plane is divided into regions marked Li (i =, 1, ) : the subscript indicates the number of limit cycles that encircle the coexistence equilibrium
3 Summary Laissez-faire & Leslie-type models with Type I functional Responses Two limit cycles: Cyclic-fold bifurcations Leslie-type model with Type I functional Responses Super-critical Hopf and Cyclic-fold bifurcations At ˆx =: the generalized Jacobian of Clarke (1998) J Σ = {(1 q)j + qj +, q 1} Discontinuous Hopf bifurcation when q = A B +1 where B A B 1 < 1
4 Future Research Projects A predator!prey model with a type I functional response including an Allee effect in the growth rate of the prey. A predator!predator!prey model with two predators characterized by type I and other possible functional responses. A Delay-differential Equation
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