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Classication of bifurcation curves for a multiparameter diffusive logistic problem with Holling type-III functional response Tzung-Shin Yeh

Department of Applied Mathematics, National University of Tainan Tainan, Taiwan 700, Republic of China

NSYSU

January 23, 2016

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 1 / 87

• 1. Introduction

We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter diffusive logistic problem with Holling type-III functional response 8 < : u00(x) + λ

• ru
• 1 u

q

• up

1+ up

• = 0, 1 < x < 1,

u(1) = u(1) = 0, (1.1) where u is the population density of the species, f (u) = ug(u) is the growth rate, g(u) = r

• 1 u

q

• up1

1+ up , (1.2) is the growth rate per capita, p > 1, q,r are two positive dimensionless parameters, and λ > 0 is a bifurcation parameter. On the right-hand side of (1.2), the rst term r(1 u

q) is the per capita birth

rate and the second term up1

1+up is the per capita death rate.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 2 / 87

8 < : u00(x) + λ

• ru
• 1 u

q

• up

1+ up

• = 0, 1 < x < 1,

u(1) = u(1) = 0, (1.1) We dene the bifurcation curve of (1.1) ¯ S = f(λ,kuλk∞) : λ > 0 and uλ is a positive solution of (1.1)g.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 3 / 87

(I) We say that the bifurcation curve ¯ S is an S-shaped curve on the (λ,jjujj∞)-plane if ¯ S consists of exactly one continuous curve with exactly two turning points at some points (λ ,kuλ k∞) and (λ ,

• uλ
• ∞) such that

(i) λ < λ and kuλ k∞ <

• uλ
• ∞,

(ii) at (λ ,kuλ k∞) the bifurcation curve ¯ S turns to the left, (iii) at (λ ,

• uλ
• ∞) the bifurcation curve ¯

S turns to the right. Note that, the upper stable branch represents outbreak states.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 4 / 87

8 < : u00(x) + λ

• ru
• 1 u

q

• up

1+ up

• = 0, 1 < x < 1,

u(1) = u(1) = 0. (1.1) (II) We say that the bifurcation curve ¯ S is a broken S-shaped curve on the (λ,jjujj∞)-plane if ¯ S has two disjoint connected components such that (i) the upper branch of ¯ S has exactly one turning point at some point (λ ,

• uλ
• ∞) where the curve turns to the right,

(ii) the lower branch of ¯ S is a monotone increasing curve.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 5 / 87

Noy-Meir studied a grazing system of herbivore-plant interaction. He considered the differential equation dN dT = G(N) Hc(N), (1.3) where N(T) is the vegetation biomass, G(N) is the growth rate of vegetation in absence of grazing, H is the herbivore population density, and c(N) is the per capita consumption rate of vegetation by the herbivore. For problem (1.3), if G(N) is given by the logistic function, and c(N) is the Holling type III function, then (1.3) takes the form dN dT = rNN

• 1 N

KN

• B

Np Ap + Np , where p > 1 and A, B, rN, KN > 0, see (Shi and Shivaji 2006).

• I. Noy-Meir, Stability of grazing systems: An application of

predator-prey graphs, J. Ecol., Vol.63 (1975), 459–481.

• J. Shi, R. Shivaji, Persistence in reaction diffusion models with weak

Allee effect, J. Math. Biol. 52 (2006) 807–829.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 6 / 87

The Holling type III functional response was also considered in (Sugie et al. 1997) and (Sugie and Katagama 1999). They studied the existence of stable limit cycle and global asymptotic stability for a predator-prey system 8 > > < > > : dx dt = rx

• 1 x

K

• xpy

Ap + xp , dy dt = y

• µxp

Ap + xp d

• .
• J. Sugie, R. Kohno, and R. Miyazaki, On a predator–prey system of

Holling type, Proc. Amer. Math. Soc., Vol.125 (1997), 2041–2050.

• J. Sugie and M. Katagama, Global asymptotic stability of a predator–prey

system of Holling type, Nonlinear Anal., Vol.38 (1999), 105–121.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 7 / 87

In addition, the Holling type III functional response also appears as the dynamics of lake eutrophication dN dT = abN + B Np Ap + Np , where N(T) is the level of nutrients suspended in phytoplankton causing turbidity, a is the nutrient loading, b is the nutrient removal rate, and B is the rate of internal nutrient recycling, see (Carpenter et al. 1999) and (Scheffer et

• al. 2001).

S.R. Carpenter, D. Ludwig, and W.A. Brock, Management of eutrophication for lakes subject to potentially irreversible change, Ecol. Appl., Vol.9 (1999), 751–771.

• M. Scheffer, S. Carpenter, J.A. Foley, C. Folke, and B. Walkerk,

Catastrophic shifts in ecosystems, Nature, Vol.413 (2001), 591–596.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 8 / 87

The model of the diffusive logistic problem with Holling type-III functional response (without diffusion)

The model of the diffusive logistic problem with Holling type-III functional response (without diffusion) is governed by the equation dN dT = rNN

• 1 N

KN

• B

Np Ap + Np , where N is the population density of the species, and

1

the rst term rNN (1N/KN) represents logistic growth, where rN is the linear birth rate of the species and KN is the carrying capacity,

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 9 / 87

The model of the diffusive logistic problem with Holling type-III functional response (without diffusion)

The model of the diffusive logistic problem with Holling type-III functional response (without diffusion) is governed by the equation dN dT = rNN

• 1 N

KN

• B

Np Ap + Np , where N is the population density of the species, and

1

the rst term rNN (1N/KN) represents logistic growth, where rN is the linear birth rate of the species and KN is the carrying capacity,

2

the second term BNp/(Ap + Np) represents predation of Holling type III functional response generated by predator, where B is a positive constant which represents the maximum predation rate of the predator and A is the species population when the predation rate is at half of the maximum, for p = 2, see (Ludwig, Jones and Holling 1978).

• D. Ludwig, D.D. Jones, C.S. Holling, Qualitative analysis of insect
• utbreak systems: the spruce budworm and forest, J. Anim. Ecol. 47

(1978) 315–332.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 9 / 87

The model of the diffusive logistic problem with Holling type-III functional response (with diffusion)

The model of the diffusive logistic problem with Holling type-III functional response (with diffusion) is governed by the equation ∂N ∂T = D∂ 2N ∂X2 + rNN

• 1 N

KN

• B

Np Ap + Np (1.4) in spatial one dimension, where D > 0 is the diffusion (dispersion) coefcient characterizing the rate of the spatial dispersion of the species population, for p = 2, see (Ludwig, Aronson and Weinberger 1979). (Note that, for the sake of simplicity, in their paper, the habitat is taken as the innite strip n (X,Y) : L

2

q

D rN < X < L 2

q

D rN , ∞ < Y < ∞

• f width L

q

D rN

and the species density is assumed to be independent of the Y coordinate.)

• D. Ludwig, D.G. Aronson, H.F. Weinberger, Spatial patterning of the

spruce budworm, J. Math. Biol. 8 (1979) 217–258.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 10 / 87

The model of the diffusive logistic problem with Holling type-III functional response (with diffusion)

∂N ∂T = D∂ 2N ∂X2 + rNN

• 1 N

KN

• B

Np Ap + Np (1.4) Let w = N A , ˜ t = rNT, ˜ x = rrN D X, r = rNA B , q = KN A . Then problem (1.4) takes the form ∂w ∂˜ t = ∂ 2w ∂ ˜ x2 + w

• 1 w

q

• 1

r wp 1+ wp . (1.5)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 11 / 87

The model of the diffusive logistic problem with Holling type-III functional response (with diffusion)

Assume that the habitat L/2 ˜ x L/2 is surrounded by a totally hostile,

• uter environment. That is, Eq. (1.5) holds in the strip j˜

xj < L/2 and w(L/2,˜ t) = w(L/2,˜ t) = 0, ˜ t > 0. (1.6) Let v(x,t) = w(˜ x,˜ t) with x = 2

L ˜

x, t = ( 2

L)2˜

t, and let λ = 1

r

L

2

• 2. Then

problem (1.5), (1.6) takes the form 8 < : ∂v ∂t = ∂ 2v ∂x2 + λ

• rv
• 1 v

q

• vp

1+ vp

• , 1 < x < 1, t > 0,

v(1,t) = v(1,t) = 0, t > 0. (1.7)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 12 / 87

8 < : ∂v ∂t = ∂ 2v ∂x2 + λ

• rv
• 1 v

q

• vp

1+ vp

• , 1 < x < 1, t > 0,

v(1,t) = v(1,t) = 0, t > 0. (1.7) Let u(x) denote a steady-state (positive) population density of (1.7). Then u(x) satises 8 < : u00(x) + λ

• ru
• 1 u

q

• up

1+ up

• = 0, 1 < x < 1,

u(1) = u(1) = 0. (1.1)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 13 / 87

Spruce budworm problem

For p = 2, problem 8 < : u00(x) + λ

• ru
• 1 u

q

• u2

1+ u2

• = 0, 1 < x < 1,

u(1) = u(1) = 0. (1.8) is a famous budworm problem in mathematical biology. For this budworm problem, roughly speaking, r measures the foliage density while q depends upon the properties of the budworm and the predators, see (Ludwig, Aronson and Weinberger 1979).

• D. Ludwig, D.G. Aronson, H.F. Weinberger, Spatial patterning of the

spruce budworm, J. Math. Biol. 8 (1979) 217–258.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 14 / 87

Some References

Problem (1.8) has been extensively studied by many authors, see (Murray 2002, 2003), (Lee, Sasi and Shivaji 2011) and (Wang and Yeh 2013). J.D. Murray, Mathematical biology. I. An introduction, Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. J.D. Murray, Mathematical biology. II. Spatial models and biomedical applications, Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.

• E. Lee, S. Sasi, and R. Shivaji, S-shaped bifurcation curves in

ecosystems, J. Math. Anal. Appl., Vol.381 (2011), 732–741. S.-H. Wang and T.-S. Yeh, S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension, J. Differential Equations, Vol.255 (2013), 812–839.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 15 / 87

Spruce budworm distribution The spruce budworm (Choristoneura fumiferana) is an very destructive native insect that lives in the spruce and r forests of Northeastern United States and Canada.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 16 / 87

Budworm Pupa Moth

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 17 / 87

Spruce tree The foliage of the spruce tree

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 18 / 87

The spruce forest Spruce budworm defoliation

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 19 / 87

Spruce budworm defoliation at British Columbia

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 20 / 87

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 21 / 87

Normally the spruce budworm exists in low numbers in these forests, kept in check by the predators (primarily birds). However, every 40 years or so there is an outbreak of these insects and their numbers can defoliate and damage most of the spruce and r trees in a forest in about 4 years. The budworm is capable of a ve-fold increase in density per year (under ideal conditions of food and weather), and the budworm can increase its density several hundred fold in a few years during outbreaks. Outbreaks can last for several years or they may collapse after only 1 or 2 years. As a consequence, the dynamics of the forest is reversed and living conditions deteriorate, but for a while the budworm density remains relatively high before it returns to low numbers again.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 22 / 87

8 < : u00(x) + λ

• ru
• 1 u

q

• u2

1+ u2

• = 0, 1 < x < 1,

u(1) = u(1) = 0. (1.8)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 23 / 87

Numerical simulation of the bifurcation curve with p=2, q=302 and r=2

S

1 2 3 4 5 50 100 150 200 250

u S

1 2 3 4 5 2 4 6 8 10

u

Numerical simulation of ¯ S for p = 2, q = 302, r = 2. (a) jjujj∞ 2 (0,250) (left); (b) jjujj∞ 2 (0,10) (right). Numerical simulation shows a big jump from point A to point B with

kvλk∞ kuλk∞ 70.661 when λ increases across λ 1.638. So an outbreak occurs

for the budworm population.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 24 / 87

Numerical simulation of the bifurcation curve with p=2, q=302 and r=0.7

S

5 10 15 20 50 100 150 200 250 300

u S

5 10 15 20 2 4 6 8 10

u

Numerical simulation of ¯ S for p = 2, q = 302, r = 0.7. (a) jjujj∞ 2 (0,300) (left); (b) jjujj∞ 2 (0,10) (right). Numerical simulation shows a huge jump from point A to point B with

kvλk∞ kuλk∞ 203.527 when λ increases across λ 11.700. So an outbreak

• ccurs hugely for the budworm population.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 25 / 87

8 < : u00(x) + λ

• ru
• 1 u

q

• u2

1+ u2

• = 0, 1 < x < 1,

u(1) = u(1) = 0. (1.8) Appropriate values for the parameters q and r in (1.8) have been estimated.

1

Based on basic ecological knowledge (Level II-general quantitative information in (Ludwig, Jones and Holling, 1978)), q ranges from 50 to 300 and r will range a minimum near 0 (for an infant forest) to a maximum of 1.07 to 3.84 (for a mature forest).

2

The studies in the more renement from extensive eld study of the forest lead to the parameters q = 302 and r ranges a minimum near 0 to a maximum of 0.994.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 26 / 87

8 < : u00(x) + λ

• ru
• 1 u

q

• u2

1+ u2

• = 0, 1 < x < 1,

u(1) = u(1) = 0. (1.8) Accordingly, (Ludwig, Aronson and Weinberger 1979) studied the exact multiplicity of positive solutions and the shape of bifurcation curve ¯ S of spruce budworm population problem (1.8) for various parameters q,r > 0. In particular, they chose parameter q = 302 and several various values of parameter r = 2, 0.7, 0.2, 0.015, 0.01. But their arguments are mostly not rigorous proofs.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 27 / 87

8 < : u00(x) + λ

• ru
• 1 u

q

• u2

1+ u2

• = 0, 1 < x < 1,

u(1) = u(1) = 0. (1.8) Applying the quadrature method (time-map method), (Ludwig, Aronson and Weinberger 1979) showed that the rough bifurcation curve goes from a monotone curve with a unique small steady state, to a broken S-shaped curve, to an S-shaped curve, and nally a monotone curve with a unique large steady state, when r increases from 0+ to a large value. Note that the results of evolutionary bifurcation curves in (Ludwig, Aronson and Weinberger 1979) are not exact, and it was only shown that the equation has at least three positive solutions but not exactly three.

• D. Ludwig, D.G. Aronson, H.F. Weinberger, Spatial patterning of the

spruce budworm, J. Math. Biol. 8 (1979) 217–258.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 28 / 87

Recently, (Wang and Yeh 2013) gave a partial answer of this conjecture in (Ludwig, Aronson and Weinberger 1979). Assume that either r ρ1q and (q,r) lies above the curve Γ =

• (q,r) : q(a) =

2a3 a2 1, r(a) = 2a3 (a2 + 1)2 , 1 < a < p 3

• r r ρ2q for some constants ρ1 0.0939 and ρ2 0.0766. Then on the

(λ,jjujj∞)-plane, they gave a classication of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Their results settled rigorously a long-standing

• pen problem in (Ludwig, Aronson and Weinberger 1979).

S.-H. Wang and T.-S. Yeh, S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension, J. Differential Equations, Vol.255 (2013), 812–839.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 29 / 87

For p > 1, the exact multiplicity of positive solutions and the shape of bifurcation curve ¯ S for (1.1) and the n-dimensional problem of (1.1) 8 < : 4u(x) + λ

• ru
• 1 u

q

• up

1+ up

• = 0, x 2 Ω,

u(x) = 0, x 2 ∂Ω remain mostly open since 1979, see (Jiang and Shi, 2009). One of the main difculties is that the growth rate per capita g(u) = r(1 u

q) up1 1+up could

initially decrease, but then increases to a peak before falling to zero.

• J. Jiang, J. Shi, Bistability dynamics in some structured ecological

models, in “Spatial Ecology”, pp. 33–61, Chapman & Hall/CRC Press, Boca Raton, FL, 2009.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 30 / 87

Note that, for p = 1, the function u 1+ u is called a Holling type-II function. A n-dimensional Dirichlet problem of (1.1) with p = 1 8 < : 4u(x) + λ

• ru
• 1 u

q

• u

1+ u

• = 0, x 2 Ω,

u(x) = 0, x 2 ∂Ω was considered by (Korman and Shi 2001). They obtained two qualitatively different bifurcation curves: a -shaped curve and a monotone increasing curve, see (Korman and Shi 2001).

• P. Korman and J. Shi, New exact multiplicity results with an application

to a population model, Proc. Royal. Soc. Edinburgh Sect. A, Vol.131 (2001), 1167–1182.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 31 / 87

• Fig. 1. Classied graphs of growth rate per capita g(u) = r(1 u

q) up1 1+up on

(0,∞), drawn on the rst quadrant of (q,r)-parameter plane.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 32 / 87

8 < : u00(x) + λ

• ru
• 1 u

q

• up

1+ up

• = 0, 1 < x < 1,

u(1) = u(1) = 0, (1.1) For problem (1.1) with xed p > 1, in Fig. 1, we divide the rst quadrant of (q,r)-parameter plane into the disjoint union of the three curves Γ0, Γ1, Γ2 and four regions R1, R2, R3, R4 dened as follows: Γ0 = f(q,r) : r = mpq > 0g, Γ1 = n (q,r) : q(a) = a[2ap(p2)]

ap(p1) , r(a) = ap1[2ap(p2)] (ap+1)2

,

p

p p1 < a < C

p

• ,

Γ2 = n (q,r) : q(a) = a[2ap(p2)]

ap(p1) , r(a) = ap1[2ap(p2)] (ap+1)2

, C

p < a < ∞

• ,

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 33 / 87

R1 = f(q,r) : 0 < r < mpq and (q,r) lies above the curve Γ1g, R2 = f(q,r) : 0 < r < mpq, and (q,r) lies between curves Γ1 and Γ2g, R3 = f(q,r) : 0 < r < mpq and (q,r) lies below the curve Γ2g, R4 = f(q,r) : r > mpq > 0g, where C

p =

• p2+3p4+pp

p2+6p7 4

1/p >

p

p p1 > 0, mp = (C

p)p2[1p+ (C p)p]

[1+ (C

p)p]2

> 0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 34 / 87

According to (Jiang and Shi, 2009), we classify all growth rate patterns according to the monotonicity of the growth rate per capita g(u) = r(1 u

q) up1 1+up on [0,∞):

1

g(u) is of logistic type, if g(u) is strictly decreasing;

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 35 / 87

According to (Jiang and Shi, 2009), we classify all growth rate patterns according to the monotonicity of the growth rate per capita g(u) = r(1 u

q) up1 1+up on [0,∞):

1

g(u) is of logistic type, if g(u) is strictly decreasing;

2

g(u) is of hysteresis type, if g(u) changes from decreasing to increasing then to decreasing again when u increases. In the hysteresis case, if g(u) has three positive zeros, then it is strong hysteresis, otherwise it is weak hysteresis.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 35 / 87

We have that: (i) If (q,r) 2 Γ0 [R4, then g is of logistic type. In this case g(u) has exactly

• ne positive zero at some β 1. The bifurcation curve ¯

S of positive solutions of (1.1) is a monotone increasing curve since f (u) uf 0(u) = ug(u) u(g(u) + ug0(u)) = u2g0(u) > 0 on (0,β 1) except possibly at some value β 0 2 (0,β 1) when (q,r) 2 Γ0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 36 / 87

We have that: (i) If (q,r) 2 Γ0 [R4, then g is of logistic type. In this case g(u) has exactly

• ne positive zero at some β 1. The bifurcation curve ¯

S of positive solutions of (1.1) is a monotone increasing curve since f (u) uf 0(u) = ug(u) u(g(u) + ug0(u)) = u2g0(u) > 0 on (0,β 1) except possibly at some value β 0 2 (0,β 1) when (q,r) 2 Γ0. (ii) If (q,r) 2 Γ2 [R3, then g is of weak hysteresis type. In this case g(u) has exactly one positive zero at some β 1. The bifurcation curve ¯ S of positive solutions of (1.1) is a monotone increasing curve since f (u) uf 0(u) = u2g0(u) > 0 on (0,β 1).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 36 / 87

(iii) If (q,r) 2 Γ1 [R2, then g is of hysteresis type. In particular, g is of strong hysteresis type if (q,r) 2 R2. Notice that:

(a) For (q,r) 2 Γ1, f (u) = ug(u) has exactly two positive zeros at some β 1 < β 3 such that g(u) > 0 on (0,β 1)[(β 1,β 3), g(β 1) = g(β 3) = 0 and g(u) < 0 on (β 3,∞). (b) For (q,r) 2 R2 with q xed, f (u) = ug(u) has exactly three positive zeros at some β 1 < β 2 < β 3 such that g(u) > 0 on (0,β 1) [(β 2,β 3), g(β 1) = g(β 2) = g(β 3) = 0 and g(u) < 0 on (β 1,β 2) [(β 3,∞).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 37 / 87

In addition, for xed q > q(p)

2

• p+1+p

p2+6p7

• p1+p

p2+6p7

• p2+3p4+pp

p2+6p7 4

1/p , there exists ¯ r2 = ¯ r2(q) 2 (r2(q),r1(q)) such that

R β 3

β 1 f (u)du < 0

for r2(q) < r < ¯ r2(q),

R β 3

β 1 f (u)du = 0

for r = ¯ r2(q),

R β 3

β 1 f (u)du > 0

for ¯ r2(q) < r < r1(q).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 38 / 87

We thus dene the curve ¯ Γ2 = f(q,r) : q > q(p) and r = ¯ r2(q)g, and regions ˆ R2 = (q,r) : 0 < r < mpq, and (q,r) lies between curves Γ1 and ¯ Γ2

• ,

¯ R2 = (q,r) : 0 < r < mpq, and (q,r) lies on curve ¯ Γ2

• r between curves Γ2 and ¯

Γ2

• .

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 39 / 87

Notice that

R β 3

β 1 f (u)du > 0 for ¯

r2(q) < r < r1(q), then there exists a number γ 2 (β 2,β 3) such that

R γ

β 1 f (u)du = 0.

(iv) If (q,r) 2 R1, then g is of weak hysteresis type. In this case g(u) has exactly one positive zero at some β 3, g(u) changes from decreasing to increasing then to decreasing on [0,β 3), g(u) > 0 on (0,β 3), g(β 3) = 0, and g(u) < 0 on (β 3,∞).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 40 / 87

• 2. Main results (Theorems 2.1, 2.2 and 2.3)
• Fig. 2. (a) S-shaped bifurcation curve ¯

S of (1.1). (b)–(c) Broken S-shaped bifurcation curves ¯ S of (1.1).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 41 / 87

For xed p > 1, we dene functions I(u) = pup2[(p1) + (p+ 1)up] 2(1+ up)3 , J(u) = up2(1p+ up) (1+ up)2 , K(u) = 4 u4 u2 2 + u2 1+ up 3

Z u

t 1+ tp dt

• ,

M(u) = 3 u3

• u+

u 1+ up 2

Z u

1 1+ tp dt

• .

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 42 / 87

I u J u K u

1,p

u

Graphs of functions η =I(u), η =J(u), η =K(u), η =η1. We rst dene two positive numbers η1,p and η2,p for p > 1 as follows: (i) Let η1,p be the unique positive intersection value of the two curves η = I(u) and η = K(u) for u > 0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 43 / 87

I u J u M u

2,p

u

Graphs of functions η =I(u), η =J(u), η =M(u), η =η2. (ii) Let η2,p be the unique positive intersection value of the two curves η = I(u) and η = M(u) for u > 0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 44 / 87

1,p 2,p

mp

1 2 3 4 5 6 7 8 9 10 p 0.1 0.2 0.3 0.4

.

Remark

We know that for p > 1, η1,p < J(C

p) = mp and η2,p < J(C p) = mp.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 45 / 87

For p > 1 and 0 < η < mp, let B1,p(η) be the smallest positive root of I(u) = η and C2,p(η) be the largest positive root of J(u) = η. We also dene two positive numbers η

1,p and η 2,p for p > 1 as follows:

(i) Let η

1,p sup

• η : 0 < η η1,p and N1(B1,p(η)) + N2(C2,p(η)) > 0
• ,

where N1(u)

u2[12+(p25p24)up+(p25p12)u2p] 4(1+up)3

+ 6

Z u

t 1+ tp dt and N2(u) u2[6+(p+7)up+u2p]

2(1+up)2

6

Z u

t 1+ tp dt.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 46 / 87

(ii) Let η

2,p sup

• η : 0 < η η2,p and N3(B1,p(η)) + N4(C2,p(η)) > 0
• ,

where N3(u) [6u(p24p12)up+1+(p2+4p+6)u2p+1]

3(1+up)3

+ 2

Z u

1 1+ tp dt and N4(u) u(6+(p+8)up+2u2p)

3(1+up)2

2

Z u

1 1+ tp dt.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 47 / 87

Remark

Numerical simulations show that, for p 2 [1.01,10], N1(B1,p(η1,p)) + N2(C2,p(η1,p)) > 0 and N3(B1,p(η2,p)) + N4(C2,p(η2,p)) > 0. Hence we obtain that, for p 2 [1.01,10], η

1,p = sup

• η : 0 < η η1,p and N1(B1,p(η)) + N2(C2,p(η)) > 0

= η1,p and η

2,p = sup

• η : 0 < η η2,p and N3(B1,p(η)) + N4(C2,p(η)) > 0

= η2,p. In particular, for p = 2, we obtain η

1,p = η1,p 0.0939 and

η

2,p = η2,p 0.0766.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 48 / 87

Let uλ be a positive solution of (1.1) with α kuλk∞ > 0.

Theorem (2.1)

Consider (1.1) with p > 1. If (q,r) 2 R1and r η

1,pq, then

limα!0+ λ(α) = ˆ λ π2

4r , limα!β

3 λ(α) = ∞, and the bifurcation curve ¯

S of (1.1) is an S-shaped curve on the (λ,jjujj∞)-plane. More precisely, ¯ S consists

• f a continuous curve with exactly two turning points at some points

(λ ,kuλ k∞) and (λ ,

• uλ
• ∞) such that ˆ

λ < λ < λ < ∞ and 0 < kuλ k∞ <

• uλ
• ∞ < β 3.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 49 / 87

Theorem (2.1)

Problem (1.1) has: (i) exactly three positive solutions wλ, uλ, vλ with wλ < uλ < vλ for λ < λ < λ , (ii) exactly two positive solutions wλ, uλ with wλ < uλ for λ = λ and exactly two positive solutions uλ, vλ with uλ < vλ for λ = λ , (iii) exactly one positive solution wλ for ˆ λ < λ < λ and exactly one positive solution vλ for λ > λ , (iv) no positive solution for 0 < λ ˆ λ.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 50 / 87

Theorem (2.1)

Furthermore, limλ!( ˆ

λ)+ kwλk∞ = 0 and limλ!∞ kvλk∞ = β 3.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 51 / 87

Theorem (2.2)

Consider (1.1) with p > 1. If (q,r) 2 ˆ R2 and r η

2,pq, then

limα!0+ λ(α) = ˆ λ π2

4r ,

limα!β

1 λ(α) = limα!γ+ λ(α) = limα!β 3 λ(α) = ∞, and the bifurcation

curve ¯ S of (1.1) is a broken S-shaped curve on the (λ,jjujj∞)-plane. More precisely, ¯ S has two disjoint connected components such that the upper branch of ¯ S has exactly one turning point (λ ,

• uλ
• ∞), with ˆ

λ < λ < ∞ and γ <

• uλ
• ∞ < β 3 , where the curve turns to the right, and the lower branch of

¯ S is a monotone increasing curve starting at ( ˆ λ,0).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 52 / 87

Theorem (2.2)

Problem (1.1) has: (i) exactly three positive solutions wλ, uλ, vλ with wλ < uλ < vλ for λ > λ , (ii) exactly two positive solutions wλ, uλ with wλ < uλ for λ = λ , (iii) exactly one positive solution wλ for ˆ λ < λ < λ , (iv) no positive solution for 0 < λ ˆ λ.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 53 / 87

Theorem (2.2)

Furthermore, lim

λ!( ˆ λ)+ kwλk∞ = 0, lim λ!∞kwλk∞ = β 1, lim λ!∞kuλk∞ = γ, and lim λ!∞kvλk∞ = β 3.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 54 / 87

Theorem (2.3)

Consider (1.1) with p > 1. If (q,r) 2 Γ1 and r η

2,pq, then

limα!0+ λ(α) = ˆ λ π2

4r ,

limα!β

1 λ(α) = limα!β + 1 λ(α) = limα!β 3 λ(α) = ∞, and the bifurcation

curve ¯ S of (1.1) is a broken S-shaped curve on the (λ,jjujj∞)-plane. More precisely, ¯ S has two disjoint connected components such that the upper branch of ¯ S has exactly one turning point (λ ,

• uλ
• ∞), with ˆ

λ < λ < ∞ and β 1 <

• uλ
• ∞ < β 3 , where the curve turns to the right, and the lower branch
• f ¯

S is a monotone increasing curve starting at ( ˆ λ,0).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 55 / 87

Theorem (2.3)

Problem (1.1) has: (i) exactly three positive solutions wλ, uλ, vλ with wλ < uλ < vλ for λ > λ , (ii) exactly two positive solutions wλ, uλ with wλ < uλ for λ = λ , (iii) exactly one positive solution wλ for ˆ λ < λ < λ , (iv) no positive solution for 0 < λ ˆ λ.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 56 / 87

Theorem (2.3)

Furthermore, lim

λ!( ˆ λ)+ kwλk∞ = 0, lim λ!∞kwλk∞ = β 1 = lim λ!∞kuλk∞ , and lim λ!∞kvλk∞ = β 3.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 57 / 87

• 3. Lemma 3.1 and the time-map method

( u00(x) + λ h ru

• 1 u

q

• up

1+up ,

i = 0, 1 < x < 1, u(1) = u(1) = 0. (1.1) For any λ > 0, let F(u)

R u

0 f (t)dt. The time map formula which we take to

prove Theorem 2.1 for problem (1.1) takes the form as follows: T(α) 1 p 2

Z α

0 [F(α) F(u)]1/2 du =

p λ for 0 < α < β 3. Positive solutions uλ of (1.1) correspond to (T(α))2 = λ and kuλk∞ = α.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 58 / 87

(T(α))2 = λ and kuλk∞ = α 2 (0,β 3). Thus, the study of the exact number of positive solutions for problem (1.1) is equivalent to that of the shape of the time map T(α) on (0,β 3).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 59 / 87

Consider the bifurcation curve of positive solutions of the problem

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1) where ˆ f 2 C[0,∞)\C2(0,∞), and λ > 0 is a bifurcation parameter. We dene ˆ F(u) =

Z u

ˆ f (t)dt, ˆ θ(u) = 2 ˆ F(u)uˆ f (u), and ˆ H(u) = 3

Z u

0 tˆ

f (t)dtu2ˆ f (u) and assume that ˆ f satises the following hypotheses (H1a) and (H2a): (H1a) There exists a number ˆ β > 0 such that ˆ f (0) = ˆ f ( ˆ β) = 0, ˆ f (u) > 0 for u 2 (0, ˆ β), and ˆ f (u) < 0 for u 2 ( ˆ β,∞). (H2a) There exist numbers 0 < ˆ B1 < ˆ C1 < ˆ B2 < ˆ C2 < ˆ β such that ˆ θ

0( ˆ

C1) = ˆ θ

0( ˆ

C2) = 0, ˆ θ

00(u) = uˆ

f 00(u) 8 < : > 0 on (0, ˆ B1) [( ˆ B2, ˆ β), = 0 for u = ˆ B1 and ˆ B2, < 0 on ( ˆ B1, ˆ B2), ˆ H( ˆ B2) 0, and 2 ˆ H( ˆ B1) ˆ B2

1 ˆ

θ

0( ˆ

B1) 2 ˆ H( ˆ C2).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 60 / 87

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1) To prove Theorem 2.1, we need the following key lemma: Lemma 3.1. We prove Lemma 3.1 by applying the time-mapping method (quadrature method) which was used by Ludwig, Aronson and Weinberger. The time map formula which we apply to study problem (3.1) takes the form as follows: T(α) 1 p 2

Z α

1 ˆ F(α) ˆ F(u) 1/2 du = p λ for 0 < α < ˆ β. So positive solutions uλ of (3.1) correspond to kuλk∞ = α and T(α) = p λ. Thus, studying the exact number of positive solutions of (3.1) is equivalent to studying the number of roots of the equation T(α) = p λ on (0, ˆ β).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 61 / 87

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1)

Lemma (3.1)

Consider (3.1). Suppose ˆ f 2 C[0,∞) \C2(0,∞) satises (H1a) and (H2a), then lim

α!0+ T(α) =

π 2pm0 2 [0,∞), lim

α! ˆ β

T(α) = ∞,

where 0 < limu!0+ ˆ f (u)/u m0 ∞, and T(α) has exactly two positive critical points, α < α, on (0, ˆ β), such that T(α) is a local maximum on (0, ˆ β) and T(α) is a local minimum on (0, ˆ β).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 62 / 87

• 3. Lemma 3.2 and the time-map method

( u00(x) + λ h ru

• 1 u

q

• up

1+up ,

i = 0, 1 < x < 1, u(1) = u(1) = 0. (1.1) For any λ > 0, let F(u)

R u

0 f (t)dt. The time map formula which we take to

prove Theorem 2.2 for problem (1.1) takes the form as follows: T(α) 1 p 2

Z α

0 [F(α) F(u)]1/2 du =

p λ for α 2 (0,β 1) [(γ,β 3). Positive solutions uλ of (1.1) correspond to (T(α))2 = λ and kuλk∞ = α.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 63 / 87

(T(α))2 = λ and kuλk∞ = α 2 (0,β 1) [(γ,β 3). Thus, the study of the exact number of positive solutions for problem (1.1) is equivalent to that of the shape of the time map T(α) on (0,β 1) [(γ,β 3).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 64 / 87

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1) We dene ˆ F(u) =

Z u

ˆ f (t)dt, ˆ θ(u) = 2 ˆ F(u)uˆ f (u), and ˆ H(u) = 3

Z u

0 tˆ

f (t)dtu2ˆ f (u) and assume that ˆ f satises the following hypotheses (H1b), (H2b) and (H3): (H1b) There exist numbers 0 < ˆ β 1 < ˆ β 2 < ˆ β such that ˆ f (0) = ˆ f ( ˆ β 1) = ˆ f ( ˆ β 2) = ˆ f ( ˆ β) = 0, ˆ f (u) > 0 on (0, ˆ β 1) [( ˆ β 2, ˆ β), and ˆ f (u) < 0 on ( ˆ β 1, ˆ β 2) [( ˆ β,∞). (H2b) There exist numbers 0 < ˆ B1 < ˆ C1 < ˆ E1 ˆ B2 < ˆ C2 < ˆ E2 < ˆ β such that ˆ θ( ˆ E1) = ˆ θ( ˆ E2) = ˆ θ

0( ˆ

C1) = ˆ θ

0( ˆ

C2) = 0 and ˆ θ

00(u) = uˆ

f 00(u) 8 < : > 0 on (0, ˆ B1) [( ˆ B2, ˆ β), = 0 for u = ˆ B1 and ˆ B2, < 0 on ( ˆ B1, ˆ B2). Also, ˆ θ( ˆ B1) ˆ B1 ˆ θ

0( ˆ

B1) ˆ θ( ˆ C2) 0. (H3) There exists a positive number ˆ γ 2 ( ˆ β 2, ˆ β) satises

R ˆ

γ ˆ β 1

ˆ f (u)du = 0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 65 / 87

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1) To prove Theorem 2.2, we need the following key lemma: Lemma 3.2. We prove Lemma 3.2 by applying the time-mapping method (quadrature method) which was used by Ludwig, Aronson and Weinberger. The time map formula which we apply to study problem (3.1) takes the form as follows: T(α) 1 p 2

Z α

1 ˆ F(α) ˆ F(u) 1/2 du = p λ for α 2 (0, ˆ β 1) [(ˆ γ, ˆ β). So positive solutions uλ of (3.1) correspond to kuλk∞ = α and T(α) = p λ. Thus, studying the exact number of positive solutions of (3.1) is equivalent to studying the number of roots of the equation T(α) = p λ on (0, ˆ β).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 66 / 87

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1)

Lemma (3.2)

Consider (3.1). Suppose ˆ f 2 C[0,∞) \C2(0,∞) satises (H1b), (H2b) and (H3), then lim

α!0+ T(α) =

π 2pm0 2 [0,∞), lim

α! ˆ β

• 1

T(α) = lim

α!ˆ γ+ T(α) = lim α! ˆ β

T(α) = ∞,

where 0 < limu!0+ ˆ f (u)/u m0 ∞, and T(α) is strictly increasing on (0, ˆ β 1) and T(α) has exactly one positive critical point at some α on (ˆ γ, ˆ β), such that T(α) is a local minimum on (ˆ γ, ˆ β).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 67 / 87

• 3. Lemma 3.3 and the time-map method

( u00(x) + λ h ru

• 1 u

q

• up

1+up ,

i = 0, 1 < x < 1, u(1) = u(1) = 0. (1.1) For any ε > 0, let F(u)

R u

0 f (t)dt. The time map formula which we take to

prove Theorem 2.3 for problem (1.1) takes the form as follows: T(α) 1 p 2

Z α

0 [F(α)F(u)]1/2 du =

p λ for α 2 (0,β 1) [(β 1,β 3). Positive solutions uλ of (1.1) correspond to (T(α))2 = λ and kuλk∞ = α.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 68 / 87

(T(α))2 = λ and kuλk∞ = α 2 (0,β 1) [(β 1,β 3). Thus, the study of the exact number of positive solutions for problem (1.1) is equivalent to that of the shape of the time map T(α) on (0,β 1) [(β 1,β 3).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 69 / 87

Consider the bifurcation curve of positive solutions of the problem

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1) where ˆ f 2 C[0,∞)\C2(0,∞), and λ > 0 is a bifurcation parameter. We dene ˆ F(u) =

Z u

ˆ f (t)dt, ˆ θ(u) = 2 ˆ F(u)uˆ f (u), and ˆ H(u) = 3

Z u

0 tˆ

f (t)dtu2ˆ f (u) and assume that ˆ f satises the following hypotheses hypotheses (H1c) and (H2c): (H1c) There exist numbers 0 < ˆ β 1 < ˆ β such that ˆ f (0) = ˆ f ( ˆ β 1) = ˆ f ( ˆ β) = 0, and ˆ f (u) > 0 for u 2 (0, ˆ β 1) [( ˆ β 1, ˆ β), and ˆ f (u) < 0 for u 2 ( ˆ β,∞). (H2c) There exist numbers 0 < ˆ B1 < ˆ C1 < ˆ E1 ˆ B2 < ˆ C2 < ˆ E2 < ˆ β such that ˆ θ( ˆ E1) = ˆ θ( ˆ E2) = ˆ θ

0( ˆ

C1) = ˆ θ

0( ˆ

C2) = 0 and ˆ θ

00(u) = uˆ

f 00(u) 8 < : > 0 on (0, ˆ B1) [( ˆ B2, ˆ β), = 0 for u = ˆ B1 and u = ˆ B2, < 0 on ( ˆ B1, ˆ B2). Also, ˆ θ( ˆ B1) ˆ B1 ˆ θ

0( ˆ

B1) ˆ θ( ˆ C2) 0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 70 / 87

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1) To prove Theorem 2.3, we need the following key lemma: Lemma 3.3. We prove Lemma 3.3 by applying the time-mapping method (quadrature method) which was used by Ludwig, Aronson and Weinberger. The time map formula which we apply to study problem (3.1) takes the form as follows: T(α) 1 p 2

Z α

1 ˆ F(α) ˆ F(u) 1/2 du = p λ for α 2 (0, ˆ β 1) [(ˆ γ, ˆ β). So positive solutions uλ of (3.1) correspond to kuλk∞ = α and T(α) = p λ. Thus, studying the exact number of positive solutions of (3.1) is equivalent to studying the number of roots of the equation T(α) = p λ on (0, ˆ β).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 71 / 87

• u00(x) + λ ˆ

f (u(x)) = 0, 1 < x < 1, u(1) = u(1) = 0, (3.1)

Lemma (3.3)

Consider (3.1). Suppose ˆ f 2 C[0,∞) \C2(0,∞) satises (H1c) and (H2c), then lim

α!0+ T(α) =

π 2pm0 2 [0,∞), lim

α! ˆ β

• 1

T(α) = lim

α! ˆ β

+ 1

T(α) = lim

α! ˆ β

T(α) = ∞,

where 0 < limu!0+ ˆ f (u)/u m0 ∞, and T(α) is strictly increasing on (0, ˆ β 1) and T(α) has exactly one positive critical point at some α on ( ˆ β 1, ˆ β), such that T(α) is a local minimum on ( ˆ β 1, ˆ β).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 72 / 87

Thanks for your attention

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 73 / 87

• 4. The Time-mapping method (quadrature method) and

derivation of the time-map formula for a two-point boundary value problem

Consider

• u00(x) + λf (u(x)) = 0, 1 < x < 1,

u(1) = u(1) = 0. (7.1) The time-map formula which we apply to study (7.1) takes the form as follows: T(α) 1 p 2

Z α

1 [F(α) F(u)]1/2 du = p λ for 0 < α < β for some β ∞. (7.2) On the (λ,kuk∞)-plane, we dene the bifurcation curve of (7.1) S = f(λ,kuλk∞) : λ > 0 and uλ is a positive solution of (7.1)g.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 74 / 87

• u00(x) + λf (u(x)) = 0, 1 < x < 1,

u(1) = u(1) = 0. (7.1) T(α) 1 p 2

Z α

1 [F(α) F(u)]1/2 du = p λ for 0 < α < β. (7.2) It can be easily proved that positive solutions uλ of (7.1) correspond to kuλk∞ = α and T(α) = p λ. Thus to study the number of positive solutions of (7.1) is equivalent to study the shape of the time map T(α) on (0,β).

2

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 75 / 87

• u00(x) + λf (u(x)) = 0, 1 < x < 1,

u(1) = u(1) = 0. (7.1) Derivation of the time-map formula T(α). T(α) 1 p 2

Z α

1 [F(α) F(u)]1/p du = p λ for 0 < α < β. (7.2)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 76 / 87

• u00(x) + λf (u(x)) = 0, 1 < x < 1,

u(1) = u(1) = 0. (7.1) Derivation of the time-map formula T(α). T(α) 1 p 2

Z α

1 [F(α) F(u)]1/p du = p λ for 0 < α < β. (7.2) Multiplying (7.1) by u0(x) and integrating, we obtain (u0(x))2 2 + λF(u(x)) = constant, 1 < x < 1, (7.3) where F(u) =

R u

0 f (t)dt.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 76 / 87

Assume x0 2 (1,1) be a maximum point of u(x) and x 2 (1,1). By (7.3), then we have u0(x) = p 2λ [F(u(x0))F(u(x))]1/2 sign(x0 x), for 1 < x < 1.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 77 / 87

Assume x0 2 (1,1) be a maximum point of u(x) and x 2 (1,1). By (7.3), then we have u0(x) = p 2λ [F(u(x0))F(u(x))]1/2 sign(x0 x), for 1 < x < 1. This implies that x+ 1 = 1 p 2λ

Z u(x)

[F(kuk∞)F(w)]1/2 dw, for 1 < x x0 and 1x = 1 p 2λ

Z u(x)

[F(kuk∞) F(w)]1/2 dw, for x0 x < 1.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 77 / 87

Assume x0 2 (1,1) be a maximum point of u(x) and x 2 (1,1). By (7.3), then we have u0(x) = p 2λ [F(u(x0))F(u(x))]1/2 sign(x0 x), for 1 < x < 1. This implies that x+ 1 = 1 p 2λ

Z u(x)

[F(kuk∞)F(w)]1/2 dw, for 1 < x x0 and 1x = 1 p 2λ

Z u(x)

[F(kuk∞) F(w)]1/2 dw, for x0 x < 1. Setting x = x0, we have x0 = 0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 77 / 87

Assume x0 2 (1,1) be a maximum point of u(x) and x 2 (1,1). By (7.3), then we have u0(x) = p 2λ [F(u(x0))F(u(x))]1/2 sign(x0 x), for 1 < x < 1. This implies that x+ 1 = 1 p 2λ

Z u(x)

[F(kuk∞)F(w)]1/2 dw, for 1 < x x0 and 1x = 1 p 2λ

Z u(x)

[F(kuk∞) F(w)]1/2 dw, for x0 x < 1. Setting x = x0, we have x0 = 0.This implies that u(x) has to be symmetric with respect to x = 0, and u0(x) > 0 for 1 < x < 0.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 77 / 87

Assume x0 2 (1,1) be a maximum point of u(x) and x 2 (1,1). By (7.3), then we have u0(x) = p 2λ [F(u(x0))F(u(x))]1/2 sign(x0 x), for 1 < x < 1. This implies that x+ 1 = 1 p 2λ

Z u(x)

[F(kuk∞)F(w)]1/2 dw, for 1 < x x0 and 1x = 1 p 2λ

Z u(x)

[F(kuk∞) F(w)]1/2 dw, for x0 x < 1. Setting x = x0, we have x0 = 0.This implies that u(x) has to be symmetric with respect to x = 0, and u0(x) > 0 for 1 < x < 0. Thus kuk∞ = u(0).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 77 / 87

• u00(x) + λf (u(x)) = 0, 1 < x < 1,

u(1) = u(1) = 0. (7.1) (u0(x))2 2 + λF(u(x)) = constant, 1 < x < 1, (7.3) Let α = u(0) and substitute x = 0 into (7.3), then (u0(x))2 2 + λF(u(x)) = (u0(0))2 2 + λF(u(0)) = λF(α).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 78 / 87

• u00(x) + λf (u(x)) = 0, 1 < x < 1,

u(1) = u(1) = 0. (7.1) (u0(x))2 2 + λF(u(x)) = constant, 1 < x < 1, (7.3) Let α = u(0) and substitute x = 0 into (7.3), then (u0(x))2 2 + λF(u(x)) = (u0(0))2 2 + λF(u(0)) = λF(α). Hence du dx = u0(x) = p 2λ [F(α) F(u)]1/2 , 1 < x < 0. (7.4)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 78 / 87

• u00(x) + λf (u(x)) = 0, 1 < x < 1,

u(1) = u(1) = 0. (7.1) (u0(x))2 2 + λF(u(x)) = constant, 1 < x < 1, (7.3) Let α = u(0) and substitute x = 0 into (7.3), then (u0(x))2 2 + λF(u(x)) = (u0(0))2 2 + λF(u(0)) = λF(α). Hence du dx = u0(x) = p 2λ [F(α) F(u)]1/2 , 1 < x < 0. (7.4) Now integrating (7.4) on [1,0], we obtain p λ = 1 p 2

Z α

0 [F(α) F(u)]1/2 du = T(α) for 0 < α < β.

(7.2)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 78 / 87

• 5. Holling type I-IV functions

8 < : u00(x) + λ

• ru
• 1 u

q

• p(u)
• = 0, 1 < x < 1,

u(1) = u(1) = 0. (*) Holling type-I function p(u) = p1(u) =

• u,

0 u < k/a, k, u k/a, where k,a > 0. Holling type-II function p(u) = p2(u) = u 1+ u. Holling type-IV function p(u) = p4(u) = u 1+ u2 .

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 79 / 87

Classied bifurcation curves of () with p(u) = u 1+ u2 , drawn on the rst quadrant of (q,r)-parameter plane.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 80 / 87

• 6. Dynamical behavior of the reaction-diffusion equation in
• ne space variable with

(q,r) 2 R

1 f(q,r) : (q,r) 2 R1 and r η1qg 8 < :

∂u ∂t = u00 + λf (u), 1 < x < 1, t > 0,

u(1,t) = u(1,t) = 0, t > 0, u(x,0) = u0(x) 0, 1 < x < 1. (12.1) where f (u) = ru

• 1 u

q

• u2

1+ u2 . (12.2)

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 81 / 87

References

• J. Shi, R. Shivaji, Persistence in reaction diffusion models with weak

Allee effect (Theorems 6 and 7), J. Math. Biol., 2006.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 82 / 87

Let ˜ C = fu 2 C([1,1]) : u(x) 0, 1 < x < 1; u(1) = u(1) = 0g. Then

• 1. If 0 < λ ˆ

λ, then limt!∞ u(x,t) = 0 (which is the trivial steady state solution) uniformly for 1 x 1, for all nontrivial initial conditions u0 2 ˜ C.

• 2. If ˆ

λ < λ < λ , then limt!∞ u(x,t) = u1(λ,x) (which is the unique positive steady state solution) uniformly for 1 x 1, for all nontrivial initial conditions u0 2 ˜ C.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 83 / 87

• 3. If λ = λ , then there exists a co-dimension one manifold M ˜

C such that ˜ C nM has exactly two connected components X1 and X2, such that (i) if u0 2 X1 [M, then limt!∞ u(x,t) = u2(λ ,x) which is the unique maximal positive steady state solution; (ii) if u0 2 X2, then limt!∞ u(x,t) = u1(λ ,x) which is the unique minimal positive steady state solution.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 84 / 87

• 4. If λ < λ < λ , then there exists a co-dimension one manifold M ˜

C such that ˜ C nM has exactly two connected components X1 and X2, such that (i) if u0 2 X1, then limt!∞ u(x,t) = u3(λ,x) which is the unique maximal positive steady state solution; (ii) if u0 2 X2, then limt!∞ u(x,t) = u1(λ,x) which is the unique minimal positive steady state solution; (iii) if u0 2 M, then limt!∞ u(x,t) = u2(λ,x) which is the unique non-maximal, non-minimal positive steady state solution.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 85 / 87

• 5. If λ = λ , then there exists a co-dimension one manifold M ˜

C such that ˜ C nM has exactly two connected components X1 and X2, such that (i) if u0 2 X1, then limt!∞ u(x,t) = u3(λ ,x) which is the unique maximal positive steady state solution; (ii) if u0 2 X2 [M, then limt!∞ u(x,t) = u2(λ ,x) which is the unique minimal positive steady state solution.

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 86 / 87

• 6. If λ > λ , then limt!∞ u(x,t) = u3(λ,x) (which is the unique positive

steady state solution) uniformly for 1 x 1, for all nontrivial initial populations u0(x).

Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 87 / 87