Classi�cation 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 u p 1 � u < u 00 ( x ) + λ ru = 0, � 1 < x < 1, � q 1 + u p (1.1) : u ( � 1 ) = u ( 1 ) = 0, where u is the population density of the species, f ( u ) = ug ( u ) is the growth rate, � � � u p � 1 1 � u g ( u ) = r 1 + u p , (1.2) q 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 u p � 1 1 + u p is the per capita death rate. Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 2 / 87
� � � � 8 u p 1 � u < u 00 ( x ) + λ ru = 0, � 1 < x < 1, � q 1 + u p (1.1) : u ( � 1 ) = u ( 1 ) = 0, We de�ne the bifurcation curve of (1.1) S = f ( λ , k u λ k ∞ ) : λ > 0 and u λ 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 , jj u jj ∞ ) -plane if ¯ S consists of exactly one continuous curve with exactly � � � u λ � � two turning points at some points ( λ � , k u λ � k ∞ ) and ( λ � , ∞ ) such that � � (i) λ � < λ � and k u λ � k ∞ < � u λ � � ∞ , (ii) at ( λ � , k u λ � k ∞ ) the bifurcation curve ¯ S turns to the left , � � � u λ � � ∞ ) the bifurcation curve ¯ (iii) at ( λ � , 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 � � � � u p 1 � u < u 00 ( x ) + λ ru = 0, � 1 < x < 1, � 1 + u p (1.1) q : u ( � 1 ) = u ( 1 ) = 0. (II) We say that the bifurcation curve ¯ S is a broken S-shaped curve on the ( λ , jj u jj ∞ ) -plane if ¯ S has two disjoint connected components such that (i) the upper branch of ¯ S has exactly one turning point at some point � � � u λ � � ( λ � , ∞ ) 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 � � N p dN 1 � N dT = r N N � B A p + N p , K N where p > 1 and A , B , r N , K N > 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 � � x p y dx 1 � x > > dt = rx A p + x p , < � K � � µ x p dy > > dt = y A p + x p � 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 N p dN dT = a � bN + B A p + N p , 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 � � N p dN 1 � N dT = r N N � B A p + N p , K N where N is the population density of the species, and the �rst term r N N ( 1 � N / K N ) represents logistic growth, where r N is the 1 linear birth rate of the species and K N 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 � � N p dN 1 � N dT = r N N � B A p + N p , K N where N is the population density of the species, and the �rst term r N N ( 1 � N / K N ) represents logistic growth, where r N is the 1 linear birth rate of the species and K N is the carrying capacity, the second term BN p / ( A p + N p ) represents predation of Holling type III 2 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 outbreak 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 � � ∂ T = D ∂ 2 N N p ∂ N 1 � N ∂ X 2 + r N N � B (1.4) A p + N p K N in spatial one dimension, where D > 0 is the diffusion (dispersion) coef�cient 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 n q q o q ( X , Y ) : � L r N < X < L D r N , � ∞ < Y < ∞ D D in�nite strip of width L 2 2 r N 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) � � ∂ T = D ∂ 2 N N p ∂ N 1 � N ∂ X 2 + r N N � B (1.4) A p + N p K N Let r r N w = N D X , r = r N A B , q = K N A , ˜ t = r N T , ˜ x = A . Then problem (1.4) takes the form � � t = ∂ 2 w w p ∂ w 1 � w � 1 x 2 + w 1 + w p . (1.5) q r ∂ ˜ ∂ ˜ 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, outer environment. That is, Eq. (1.5) holds in the strip j ˜ x j < L / 2 and w ( � L / 2, ˜ t ) = w ( L / 2, ˜ t ) = 0, ˜ t > 0. (1.6) � L � 2 . Then t ) with x = 2 x , t = ( 2 L ) 2 ˜ t , and let λ = 1 Let v ( x , t ) = w ( ˜ x , ˜ L ˜ r 2 problem (1.5), (1.6) takes the form � � � � 8 ∂ t = ∂ 2 v v p ∂ v 1 � v < rv , � 1 < x < 1, t > 0, ∂ x 2 + λ � 1 + v p q (1.7) : v ( � 1, t ) = v ( 1, t ) = 0, t > 0. Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 12 / 87
8 � � � � ∂ t = ∂ 2 v v p ∂ v 1 � v < rv , � 1 < x < 1, t > 0, ∂ x 2 + λ � q 1 + v p (1.7) : v ( � 1, t ) = v ( 1, t ) = 0, t > 0. Let u ( x ) denote a steady-state (positive) population density of (1.7). Then u ( x ) satis�es 8 � � � � u p 1 � u < u 00 ( x ) + λ ru = 0, � 1 < x < 1, � (1.1) q 1 + u p : u ( � 1 ) = u ( 1 ) = 0. Tzung-Shin Yeh (NUTN) A multiparameter diffusive logistic problem 13 / 87
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