A Comparison of Two Predator-Prey Models with Holling’s Type I Functional Response

** Joint work with Mark Kot at the University of Washington **

Presented by Gunog Seo

York University / Ryerson University

Mathematical Biosciences 212 (2008) 161-179

Model Formulation

dN

dT= rN

�1− N

K

�− Φ(N)P

dP

dT= G(N,P )P

P (T ) = Predator population

N(T ) = Prey population

intrinsic rate of growth

of the prey

Carrying capacity

Functional

Response

Numerical

Response

2

Model Formulation

3

Functional Responses: identified by C. S. Holling (1959, 1965, 1966)

7

a 2a

c/2

c

Type I

N

!(N

)

(a)

a 2a

c/2

c

Type II

N

!(N

)

(b)

a 2a

c/2

c

Type III

N

!(N

)

(c)

Figure 1.2: The three types of functional responses described by C.S. Holling: (a) type I,(b) type II, and (c) type III functional responses. The abscissa and ordinate represent theprey density and the number of prey eaten per predator per unit time. c is a maximum percapita consumption rate and a is a half-saturation constant.

7

a 2a

c/2

c

Type I

N

!(N

)

(a)

a 2a

c/2

c

Type II

N

!(N

)(b)

a 2a

c/2

c

Type III

N

!(N

)

(c)

Figure 1.2: The three types of functional responses described by C.S. Holling: (a) type I,(b) type II, and (c) type III functional responses. The abscissa and ordinate represent theprey density and the number of prey eaten per predator per unit time. c is a maximum percapita consumption rate and a is a half-saturation constant.

Φ(N) =cN

a+NΦ(N) =

cN2

a+N2Φ(N) =

�c2aN if N < 2a

c if N ≥ 2a.

dN

dT= rN

�1− N

K

�− Φ(N)P

dP

dT= G(N,P )P

4

Numerical Responses

Laissez-faire

Leslie

Model Formulation

dN

dT= rN

�1− N

K

�− Φ(N)P

dP

dT= G(N,P )P

G(N,P ) =b

cΦ(N)−m

G(N,P ) = s

�1− P

hN

�

Outline

Laissez-faire Model

Leslie-type Model

dN

dT= rN

�1− N

K

�− ΦI(N)P

dP

dT= P

�b

cΦI(N)−m

�

dP

dT= sP

�1− P

hN

�

dN

dT= rN

�1− N

K

�− ΦI(N)P

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.

Laissez-faire and Leslie-type models with arctan functional responses.

Discussion

!I(N) =

!

c

2aN, N < 2a,

c, N ! 2a.

if time permits

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 10th European Symposium on Marine Biology, pp. 211!229, Universa Press, Ostend, 1976.

Y. Ren and L. Han. The predator prey model with two limit cycles. Acta Math. Appl. Sinica (English Ser.), 5: 30!32, 1989.

J. B. Collings. The effects of the functional response on the bifurcation behavior of a mite predator!prey inte- raction 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!210, 1998.

X. Y. Li and W. D. Wang. Qualitative analysis of predator!prey system with Holling type I functional respon- se. J. South China Normal Univ. (Natur. Sci. Ed.), 29: 712!717, 2004.

B. Liu, Y. Zhang, and L. Chen. Dynamics complexities of a Holling I predator!prey model concerning perio- dic biological and chemical control. Chaos Solitons Fractals, 22: 123!134, 2004.

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, 2005.

6

Models with Type I Functional Responses

Laissez-faire Modelwith a type I functional response

dP

dT= P

�b

cΦI(N)−m

�

dN

dT= rN

�1− N

K

�− ΦI(N)P

Laissez-faire Model with a type I functional response

Nondimensionalization & Equilibria

8

dN

dT= rN

�1− N

K

�− ΦI(N)P

dP

dT= P

�b

cΦI(N)−m

�

where ΦI(N) =

c

2aN if N < 2a

c if N ≥ 2a

x =N

a, y =

c

raP, t = rT

α =b

r, β =

m

b, γ =

K

a

dx

dt= x

�1− x

γ

�− φI(x)y

dy

dt= αy(φI(x)− β)

where φI(x) =

�x/2, x < 2

1, x ≥ 2

Assuming α > 0, 0 < β < 1, and γ > 2

Eq

uilib

ria E0 = (0, 0),

E1 = (γ, 0),

E2 = (2β, g(2β)) where g(x) =x

φI(x)

�1− x

γ

�

Laissez-faire Model with a type I functional response

Linearized Stability Analysis

9

Assuming α > 0, 0 < β < 1, and γ > 2

Eq

uilib

ria

E0 = (0, 0),

E1 = (γ, 0),

E2 = (2β, g(2β)) where g(x) =x

φI(x)

�1− x

γ

�

J =

�φ�I(x) (g(x)− y) + φI(x)g�(x) −φI(x)

αyφ�I(x) α (φI(x)− β)

�

saddle point

saddle point if φI(γ) > β

stable node if φI(γ) < β

Laissez-faire Model with a type I functional response

Linearized Stability Analysis

10

Assuming α > 0, 0 < β < 1, and γ > 2

Eq

uilib

ria

E0 = (0, 0),

E1 = (γ, 0),

E2 = (2β, g(2β)) where g(x) =x

φI(x)

�1− x

γ

�

saddle point

saddle point if φI(γ) > β

stable node if φI(γ) < β

JE2 =

�βg�(2β) −β

αg(2β)φ�I(2β) 0

�

By Routh-Hurwitz criterion,

the coexistence equilibrium is asymptotically stable.

λ2 − βg�(2β)λ+ αβg(2β)φ�I(2β) = 0

Characteristic Equation

Laissez-faire Model with a type I functional response

Global Stability Analysis of E2

11

Using Harrison’s “Gedankenexperiment” (1979), construct Lyapunov function

V (x, y) =

� x

x∗

α(φI(ξ)− β)

φI(ξ)dξ + α

� g(x∗)

y

g(x∗)− ξ

αξdξ

where g(x) =x

φI(x)

�1− x

γ

�and x∗ = 2β

V (x, y) = V1(x) + V2(x) where V1(x) = α

�(x− x∗)− x∗ ln x

x∗ x < 2,

(2− x∗)− x∗ ln 2x∗ + (1− β)(x− 2), x ≥ 2

V2(x) = g(x∗) lng(x∗)

y+ y − g(x∗)

V

�=

dV

dt

�= α (φI(x)− β) (g(x)− g(x∗)) is continuous at x = 2

V (x, y) is continuous at x = 2 and is zero at the coexistence equilibrium

For all positive x and y, V (x, y) is positive, except at coexistence equilibrium E2.

V < 0 in a neighborhood of E2 if g(x) > g(x∗) for x < x∗ AND g(x) < g(x∗) for x > x∗.

40

0 20

2

x

y

(a)

x! !

0 20

2

x

y

(b)

x! !

D

ULC

SLC

Figure 2.2: A figure illustrating (a) isoclines consistent with global asymptotic stability and(b) a subset of the domain of attraction (D) for an equilibrium where the prey and predatorcoexist in laissez-faire model (2.6). ULC and SLC signify unstable and stable limit cycles.

Laissez-faire Model with a type I functional response

Global Stability Analysis of E2

The coexistence equilibriumis globally asymptotically stable. A subset of the basin

of attraction

D = {(x, y) | V (x, y) < u}

where

u = min {V (0, g(x!)), V (xM , g(x!)), V (x!, 0), V (x!,+!)}

with xM =! !

!

!2! 8(! ! 2")

2when ! > 4 + 4

!

1 ! ".

12

Laissez-faire Model with a type I functional response

Numerical Studies

Poincaré or first return map

(2!, g(2!))

x0

P(x0) = x

1

Poincar e section

: y = 2!1 ! 2!

"

"for x > 2!

13

Laissez-faire Model with a type I functional response

Numerical Studies

Poincaré or first return map44

2 4 6 8

1

2

3

4

5

6

7

8

x0

P(x

0)

(a)

2 4 6 8

2

3

4

5

6

7

8

x0

P(x

0)

(b)

2 4 6 8

2

3

4

5

6

7

8

x0

P(x

0)

(c)

Figure 2.5: Poincare (or first-return) maps for laissez-faire model (2.6) for ! = 2, " = 8,and for three di!erent values of #: (a) # = 0.25, (b) # = 0.65, (c) # = 0.85. Fixed points ofthe Poincare map, which correspond to either equilibrium points or limit cycles of the flow,can be found at the intersections of the Poincare map (dotted curve) and the diagonal line.No limit cycles are found for small #, but two limit cycles occur as # increases.

! = 0.25

α = 2, γ = 8

44

2 4 6 8

1

2

3

4

5

6

7

8

x0

P(x

0)

(a)

2 4 6 8

2

3

4

5

6

7

8

x0

P(x

0)

(b)

2 4 6 8

2

3

4

5

6

7

8

x0

P(x

0)

(c)

Figure 2.5: Poincare (or first-return) maps for laissez-faire model (2.6) for ! = 2, " = 8,and for three di!erent values of #: (a) # = 0.25, (b) # = 0.65, (c) # = 0.85. Fixed points ofthe Poincare map, which correspond to either equilibrium points or limit cycles of the flow,can be found at the intersections of the Poincare map (dotted curve) and the diagonal line.No limit cycles are found for small #, but two limit cycles occur as # increases.

! = 0.65

44

2 4 6 8

1

2

3

4

5

6

7

8

x0

P(x

0)

(a)

2 4 6 8

2

3

4

5

6

7

8

x0

P(x

0)

(b)

2 4 6 8

2

3

4

5

6

7

8

x0

P(x

0)

(c)

Figure 2.5: Poincare (or first-return) maps for laissez-faire model (2.6) for ! = 2, " = 8,and for three di!erent values of #: (a) # = 0.25, (b) # = 0.65, (c) # = 0.85. Fixed points ofthe Poincare map, which correspond to either equilibrium points or limit cycles of the flow,can be found at the intersections of the Poincare map (dotted curve) and the diagonal line.No limit cycles are found for small #, but two limit cycles occur as # increases.

! = 0.85

14

Laissez-faire Model with a type I functional response

Numerical Studies Bifurcation Diagrams (using XPPAUT)

46

0.5 0.6 0.7 0.8 0.9

2

3

4

5

6

(b)

!

Norm

CFB

6 8 10 12 14 16

2

4

6

8

10

(c)

"

Norm

CFB

0 2 4 6 8 10 122

2.5

3

3.5

4

4.5

5

(a)

#

Norm

CFB

CFB

Figure 2.6: Bifurcation diagrams for laissez-faire model (2.6). The L2 norms of stable limitcycles (filled circles), unstable limit cycles (open circles), and stable coexistence equilibria(thick solid line) are computed for di!erent values of (a) ! (" = 0.7, # = 8), (b) " (! = 2,# = 8), and (c) # (! = 2, " = 0.7). Two global cyclic-fold bifurcations occur as I increase(a) ! and one cyclic-fold bifurcation appears as I increase (b) " or (c) #.

46

0.5 0.6 0.7 0.8 0.9

2

3

4

5

6

(b)

!

Norm

CFB

6 8 10 12 14 16

2

4

6

8

10

(c)

"

Norm

CFB

0 2 4 6 8 10 122

2.5

3

3.5

4

4.5

5

(a)

#

Norm

CFB

CFB

Figure 2.6: Bifurcation diagrams for laissez-faire model (2.6). The L2 norms of stable limitcycles (filled circles), unstable limit cycles (open circles), and stable coexistence equilibria(thick solid line) are computed for di!erent values of (a) ! (" = 0.7, # = 8), (b) " (! = 2,# = 8), and (c) # (! = 2, " = 0.7). Two global cyclic-fold bifurcations occur as I increase(a) ! and one cyclic-fold bifurcation appears as I increase (b) " or (c) #.

46

0.5 0.6 0.7 0.8 0.9

2

3

4

5

6

(b)

!

Norm

CFB

6 8 10 12 14 16

2

4

6

8

10

(c)

"

Norm

CFB

0 2 4 6 8 10 122

2.5

3

3.5

4

4.5

5

(a)

#

Norm

CFB

CFB

Figure 2.6: Bifurcation diagrams for laissez-faire model (2.6). The L2 norms of stable limitcycles (filled circles), unstable limit cycles (open circles), and stable coexistence equilibria(thick solid line) are computed for di!erent values of (a) ! (" = 0.7, # = 8), (b) " (! = 2,# = 8), and (c) # (! = 2, " = 0.7). Two global cyclic-fold bifurcations occur as I increase(a) ! and one cyclic-fold bifurcation appears as I increase (b) " or (c) #.

47

0.1 0.2 0.3 0.4 0.5 0.6 0.7

7.8

8

8.2

8.4

8.6

8.8

9 ! = 2! = 0.5

! = 7

(a)

!

"

0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

"=

8

"=

8.5

"=

7.7

(b)

!

#

7.8 7.85 7.9 7.95 8 8.05 8.1 8.150

2

4

6

8

10

#=

0 .68

#=

0 .7

#=

0.7

2

(c)

"

#

Figure 2.7: Two-parameter bifurcation diagrams for laissez-faire model (2.6). The curvesindicate cyclic-fold bifurcations. Two limit cycles occur above each curve in (a) and to theright of each curve in (b) and (c).

47

0.1 0.2 0.3 0.4 0.5 0.6 0.7

7.8

8

8.2

8.4

8.6

8.8

9 ! = 2! = 0.5

! = 7

(a)

!

"

0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10"

=8

"=

8.5

"=

7.7

(b)

!

#

7.8 7.85 7.9 7.95 8 8.05 8.1 8.150

2

4

6

8

10

#=

0 .68

#=

0 .7

#=

0.7

2

(c)

"

#

Figure 2.7: Two-parameter bifurcation diagrams for laissez-faire model (2.6). The curvesindicate cyclic-fold bifurcations. Two limit cycles occur above each curve in (a) and to theright of each curve in (b) and (c).

47

0.1 0.2 0.3 0.4 0.5 0.6 0.7

7.8

8

8.2

8.4

8.6

8.8

9 ! = 2! = 0.5

! = 7

(a)

!

"

0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

"=

8

"=

8.5

"=

7.7

(b)

!

#

7.8 7.85 7.9 7.95 8 8.05 8.1 8.150

2

4

6

8

10

#=

0 .68

#=

0 .7

#=

0.7

2

(c)

"

#

Figure 2.7: Two-parameter bifurcation diagrams for laissez-faire model (2.6). The curvesindicate cyclic-fold bifurcations. Two limit cycles occur above each curve in (a) and to theright of each curve in (b) and (c).

! = 0.7, " = 8 ! = 2, " = 8 ! = 2, " = 0.7

Two limit cycles occur

above each curve.

Two limit cycles occur

to the right of each curve.

Two limit cycles occur

to the right of each curve.

15

Leslie-type Modelwith a type I functional response

dP

dT= sP

�1− P

hN

�

dN

dT= rN

�1− N

K

�− ΦI(N)P

Leslie-type Model with a type I functional response

Nondimensionalization & Equilibria

17

dN

dT= rN

�1− N

K

�− ΦI(N)P

where ΦI(N) =

c

2aN if N < 2a

c if N ≥ 2a

x =N

a, y =

c

raP, t = rT

dx

dt= x

�1− x

γ

�− φI(x)y

where φI(x) =

�x/2, x < 2

1, x ≥ 2

Assuming

dP

dT= sP

�1− P

hN

�

A =s

r, B =

ch

r, γ =

K

a

dy

dt= Ay

�1− y

Bx

�

A > 0, B > 0, and γ > 2

focus on the dynamics in 0 < x(t) ≤ γ , where a unique coexistence equilibrium exists.

Eq

uilib

ria �E1 = (γ, 0)

�E2 = (x∗, y∗)

where x∗ =

�2γ

Bγ+2 , γ(1−B) < 2,

γ(1−B), γ(1−B) ≥ 2

y∗ = Bx∗ = g(x∗)

with g(x) =x

φI(x)

�1− x

γ

�

Leslie-type Model with a type I functional response

Stability Analysis

18

dx

dt= x

�1− x

γ

�− φI(x)y

where φI(x) =

�x/2, x < 2

1, x ≥ 2

dy

dt= Ay

�1− y

Bx

�

x! =

!

2!B!+2

, !(1 ! B) < 2,

!(1 ! B) , !(1 ! B) " 2.

Eq

uilib

ria

�E1 = (γ, 0)�E2 = (x∗, y∗)

J =

�φ�I(x)(g(x)− y) + φI(x)g�(x) −φI(x)

AB

� yx

�2A�1− 2y

Bx

��

saddle point

E2 = (x!, y!)!

The left (J!

) and right (J+) Jacobians evaluated at E2

where 0 < x!

< 2 or 2 < x!

< !,

J±

!

!

!

(x!,y!)=

"

!I(x!)g"±(x!) ! !I(x!)AB ! A

#

with y! = g(x!) =x!(1 ! x!/!)

"I(x!)and

!

g"#

(x!) = ! 2/!, 0 < x! < 2,

g"+(x!) = 1 ! 2x!/!, 2 < x! < !.

!2 + !(A ! "I(x!)g"±(x!)) + A"I(x

!)!

B ! g"±(x!)"

= 0> 0

For 2 < x! < !/2, Asymptotically stable where 0 < x∗ < 2 or γ/2 ≤ x∗ < γ

Unstable if A < g�(x∗)

Stable if A > g�(x∗)

Leslie-type Model with a type I functional response

Stability Analysis

0 2

x

y

E2

E1

!!/2

!"#$%&'((!)(*+#,

!"#-./0"%&'((!)(*+#,

The generalized Jacobian of Clarke (1998)

J! = {(1 ! q)J!

+ qJ+ ," 0 # q # 1}

J±

!

!

!

(x!,y!)=

"

!I(x!)g"±(x!) ! !I(x!)AB ! A

#

: a convex combination of the left and right Jacobian,

! J! =

!

J11!

" 1

AB " A

"

with J11

! = qB + (B ! 1) where 0 < B < 1 and 0 " q " 1.

Characteristic Equation: λ2 −�J11Σ −A

�λ+A(B − J11

Σ ) = 0

⇒ λ1,2 =(J11

Σ −A)± i�4AB − (J11

Σ +A)2

2

A discontinuous Hopf bifurcation (Leine and Nijmeijer; 2004) is expected

when the eigenvalues are imaginary(J11

! = A) i.e., when q =A ! B + 1

Bwhere A " 2B!1 < 1.

19

Leslie-type Model with a type I functional response

The generalized Jacobian of Clarke

The generalized Jacobian of Clarke at x!

= 2

!1 !2A = 0.1 q = 0 (!), q = 1 (!)

52

!0.5 0 0.5

!0.2

0

0.2

(a)

!0.5 0 0.5

!0.2

0

0.2

(b)

!0.5 0 0.5

!0.2

0

0.2

(c)

!0.5 0 0.5

!0.2

0

0.2

(d)

!0.5 0 0.5

!0.2

0

0.2

(e)

!0.5 0 0.5

!0.2

0

0.2

(f)

Figure 2.8: Paths of eigenvalues of generalized Jacobian (2.38) evaluated at x = 2 in Leslie-type model (2.26) for A = 0.1: (a) B = 0.28, (b) B = 0.3, (c) B = 0.43, (d) B = 0.5, (e)B = 0.71, and (f) B = 0.85. The abscissa and the ordinate are the real and the imaginaryparts of !. Arrows indicate the direction of eigenvalues that start at ! (q = 0) and thatend at ! (q = 1).

52

!0.5 0 0.5

!0.2

0

0.2

(a)

!0.5 0 0.5

!0.2

0

0.2

(b)

!0.5 0 0.5

!0.2

0

0.2

(c)

!0.5 0 0.5

!0.2

0

0.2

(d)

!0.5 0 0.5

!0.2

0

0.2

(e)

!0.5 0 0.5

!0.2

0

0.2

(f)

Figure 2.8: Paths of eigenvalues of generalized Jacobian (2.38) evaluated at x = 2 in Leslie-type model (2.26) for A = 0.1: (a) B = 0.28, (b) B = 0.3, (c) B = 0.43, (d) B = 0.5, (e)B = 0.71, and (f) B = 0.85. The abscissa and the ordinate are the real and the imaginaryparts of !. Arrows indicate the direction of eigenvalues that start at ! (q = 0) and thatend at ! (q = 1).

B = 0.43

B = 0.85

20

56

0 2 4 60

0.5

1

1.5

2

2.5

x

y

(a)

DL

0 2 4 6 80

0.5

1

1.5

2

2.5

x

y

(b)

DL

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

x

y

(c)

DL

ULC

SLC

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

x

y(d)

DL

Figure 2.11: Phase portraits showing subsets of the domain of attraction (DL) for thecoexistence equilibrium of Leslie-type model (2.26) obtained using Lyapunov function (2.43).The coexistence equilibria in (a), for A = 1, B = 2, ! = 6, and (b), for A = 1, B = 0.18,! = 9, are globally asymptotically stable. A subset of the domain of attraction ((c) and(d)) can be found when g(x) > y! for x < x! and g(x) < y! for x > x!. For (c), A = 0.05,B = 1.4, ! = 10.5, while for (d), A = 0.1, B = 0.35, ! = 10. Dashed lines are the preynull-clines and solid lines are the predator null-clines. ULC and SLC signify unstable andstable limit cycles. The dotted horizontal line is y = g(x!).

Leslie-type Model with a type I functional response

Numerical Studies

56

0 2 4 60

0.5

1

1.5

2

2.5

x

y

(a)

DL

0 2 4 6 80

0.5

1

1.5

2

2.5

x

y

(b)

DL

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

x

y

(c)

DL

ULC

SLC

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

x

y

(d)

DL

Figure 2.11: Phase portraits showing subsets of the domain of attraction (DL) for thecoexistence equilibrium of Leslie-type model (2.26) obtained using Lyapunov function (2.43).The coexistence equilibria in (a), for A = 1, B = 2, ! = 6, and (b), for A = 1, B = 0.18,! = 9, are globally asymptotically stable. A subset of the domain of attraction ((c) and(d)) can be found when g(x) > y! for x < x! and g(x) < y! for x > x!. For (c), A = 0.05,B = 1.4, ! = 10.5, while for (d), A = 0.1, B = 0.35, ! = 10. Dashed lines are the preynull-clines and solid lines are the predator null-clines. ULC and SLC signify unstable andstable limit cycles. The dotted horizontal line is y = g(x!).

A = 1, B = 2, γ = 6 A = 1, B = 0.18, γ = 9

A = 0.05, B = 1.4, γ = 10.5 A = 0.1, B = 0.35, γ = 10

21

Leslie-type Model with a type I functional response

Numerical Studies: Two-parameter bifurcation diagrams

61

0.5 1 1.5 2 2.5 3 3.5

6

8

10

12

14

16

18

(a)

B

!

L0

L0

L1

L2

H+

DHBCFB

5 10 15 20 25 300

0.2

0.4

0.6

0.8

!

A

(c)

L0

L1

L2

0.4 0.6 0.8 1 1.2 1.4 1.6

0.1

0.2

0.3

0.4

0.5

0.6

B

A

(b)

L0

L0

L1

L2

Figure 2.15: Two-parameter bifurcation curves for Leslie-type model (2.26) for (a) A = 0.1,(b) ! = 10, and (c) B = 0.9. H+ stands for a super-critical Hopf bifurcation, DHB for adiscontinuous Hopf bifurcation, and CFB for a cyclic-fold bifurcation. The parameter planeis divided into regions marked Li (i = 0, 1, 2): the subscript indicates the number of limitcycles that encircle the coexistence equilibrium E2.

61

0.5 1 1.5 2 2.5 3 3.5

6

8

10

12

14

16

18

(a)

B

!

L0

L0

L1

L2

H+

DHBCFB

5 10 15 20 25 300

0.2

0.4

0.6

0.8

!

A

(c)

L0

L1

L2

0.4 0.6 0.8 1 1.2 1.4 1.6

0.1

0.2

0.3

0.4

0.5

0.6

B

A

(b)

L0

L0

L1

L2

Figure 2.15: Two-parameter bifurcation curves for Leslie-type model (2.26) for (a) A = 0.1,(b) ! = 10, and (c) B = 0.9. H+ stands for a super-critical Hopf bifurcation, DHB for adiscontinuous Hopf bifurcation, and CFB for a cyclic-fold bifurcation. The parameter planeis divided into regions marked Li (i = 0, 1, 2): the subscript indicates the number of limitcycles that encircle the coexistence equilibrium E2.

61

0.5 1 1.5 2 2.5 3 3.5

6

8

10

12

14

16

18

(a)

B

!

L0

L0

L1

L2

H+

DHBCFB

5 10 15 20 25 300

0.2

0.4

0.6

0.8

!

A

(c)

L0

L1

L2

0.4 0.6 0.8 1 1.2 1.4 1.6

0.1

0.2

0.3

0.4

0.5

0.6

B

A(b)

L0

L0

L1

L2

Figure 2.15: Two-parameter bifurcation curves for Leslie-type model (2.26) for (a) A = 0.1,(b) ! = 10, and (c) B = 0.9. H+ stands for a super-critical Hopf bifurcation, DHB for adiscontinuous Hopf bifurcation, and CFB for a cyclic-fold bifurcation. The parameter planeis divided into regions marked Li (i = 0, 1, 2): the subscript indicates the number of limitcycles that encircle the coexistence equilibrium E2.

A = 0.1

B = 0.9

The parameter plane is divided into regions marked Li (i = 0, 1, 2) : the subscript indicates the number of limit cycles that encircle the coexistence equilibrium

γ = 10

22

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

At : the generalized Jacobian of Clarke (1998)x∗ = 2

J! = {(1 ! q)J!

+ qJ+ ," 0 # q # 1}

Discontinuous Hopf bifurcation

Super-critical Hopf and Cyclic-fold bifurcations

q =A−B + 1

BA ≤ 2B − 1 < 1when where

23

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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Thank you