数理解析研究所講究録
第 1547 巻 2007 年 34-42

34

Steady state analysis for some delay equations
中岡慎治 (Shinji Nakaoka)

静岡大学大学院理工学研究科

Graduate school of Science and Technology, Shizuoka University

ABSTRACT
We study the dynamios of size-structured population model represented
by delay equations with infinite delay. The population is split into twogroups according to their maturity which is determined by their size.
Delay equations consist of a Volterra functional equation coupled with adelay differential equation which describe the time evolution of population birth rate and the food density, respectively. In this paper, steadystate analysis for the interior equilibrium of delay equations is carriedout in order to address questions under what conditions population cycles can occur.
Key words: size-structure; resource-consumer model; delay equations with infinite

delay; state dependent delay; steady-state analysis;

1

Introduction

Individuals differ from each other in terms of size and age etc.. These physiological differences affect the vital rates such as survival, development andreproduction rate. The growth in age and size is often coupled to maturation

so that reproduction takes place only after individuals have reached a certain
age or slze. The importance of body size is related to the fact that 80% of all
species grow and develop throughout their entire life (Werner [7]). Therefore
size is one of the most important individual physiological traits which would
affect to the population-level phenomena. Practically, in particular it is nat-ural for insects which typically go through several stages during their life, it
is often the case that one distinguishes individuals into several discrete stages

35in terms of age or size. In continuous time models described by delay differ-ential equations, development or transition to the next stage is described by
the sojourn time (see Cooke et al. [1], Gourley and Kuang [5] etc.). Recently,
de Roos and Persson [6] studied a size structured population model in which

two size classes, juveniles and adults, are distinguished. The model consideredin [6] is described by a system of delay differential equations with state dependent delay. By means of numerical analysis of steady state and numericalsimulations with the EBT-method, they showed that three types of population
cycles can occur depending on the nature of competition among individuals.
In this paper, we study a mathematical model which describes the population

dynamics of size structured population of the form:$b(t)=\beta_{A}(F(t))A(t)$ ,$\frac{dF}{dt}(t)=D-\gamma_{J}(F(t))J(t)-\gamma_{A}(F(t))A(t)$$J(t)= \int_{t-\tau(t)}^{t}b(\alpha)e^{-\int_{\alpha}^{t}\mu_{J}(F(\sigma))d\sigma}d\alpha$

,

,

(DE)

,

$A(t)= \int_{-\infty}^{t-\tau(t)}b(\alpha)e^{-\int_{a}^{\alpha+\overline{\tau}(\alpha)}\mu_{J}(F(\sigma))d\sigma-\int_{\alpha+\overline{\tau}(\alpha)}^{t}\mu_{A}(F(\sigma))d\sigma}d\alpha$.

$s_{m}-s_{b}= \int_{t-\tau(t)}^{t}g(F(\sigma))d\sigma=\int^{t+\tilde{\tau}(t)}g(F(\sigma))d\sigma$

Here $b(t)$ denotes the population birth rate, while $F(t)$ denotes the food density

at time . $J(t)$ and $A(t)$ denote the populatlon size of juveniles and adults at$t$time , respectively. Two types of time delay $\tau=\tau(t)$ and$t$$\tau=\tilde{\tau}(t)$areimplicitly defined by the forth equation of (DE). Note that individuals thatwere born at time $t-\tau$ , while individuals that are born,at time mature at time
. The functions $g(F),$represent the rates for individual growth, death of juveniles
andmature at time

$t$

$t$

$\gamma_{J}(F)$$t+\tilde{\tau}$$\mu_{J}(F),$$\mu_{A}(F),$$\beta_{A}(F)$$\gamma_{A}(F)$and adults, reproduction and consumption ofjuveniles and adults, respectively.

We assume that the size-at-birth of individuals is fixed at $s_{b}.We$ further assume
that the maturation size of juveniles is also fixed at $s_{m}>s_{b}$ . $D$ is the constantrate at which food is provided in the environment.

System (DE) can be derived from a size-structured resource-consumer model
described by partial differential equations as a special case (see for example,

36[4]). Note that system (DE) includes the equations (3a) and (3b) consideredin [6] as a special case. In fact, equations (3a) and (3b) correspond to (DE) if$\gamma_{J}(F)=aF,$ $\gamma_{A}(F)=qaF,$ $g(F)=\epsilon_{g}aF,$ $\mu_{J}(F)=\mu/(aF),$$\mu_{A}(F)=\mu/(qaF)$

and $\beta_{A}(F)=\epsilon_{b}qaF$ . The purpose of this paper is to investigate under what
conditions population cycles can occur by analyzing a characteristic equation
associated with the linearized equations of system (DE) around an interior

equilibrium. The organization is as follows. In the next section, we show thecondition for the existence of an interior equilibrium of system (DE). Then we
derive a linearized equations of system (DE) around the interior equilibrium.

A characteristic equation is defined from the linearized equations. Then we
look for the existence ofa complex conjugate of pure imaginary roots for thecharacteristic equation to examine whether Hopf bifurcation occurs or not. In
the last section, we discuss our results.

2

Steady state analysis

2.1

Interior equilibrium

It follows from the fourth equation $of\cdot(DE)$ , we infer that in steady state$\tilde{\tau}=\tau=\frac{s_{m}-s_{b}}{g(F)}$

.

(2.1)

For $b\neq 0$ , the steady state version of (DE) reduces to a condition on $F$ , viz.
$\beta_{A}(F)e^{-\tau\mu_{J}(F)}\frac{1}{\mu_{A}(F)}=1$

.

(22)

The left hand side is easily interpreted as the basic reproduction number $R_{0}(F)$ .
Note that one should use (2.1) to make it into a condition involving only .
$F$$g(F),$
are smooth functions of $F$ .
andWe assume that all
For
and $g(F)$ , we further assume that $\beta_{A}’(F)>0$ and $g’(F)>0$ for all.
$F\in[0, \infty)$ . While for, we further assume that $\mu_{J}’(F)\leq 0$
and$\beta_{A}(F),$$\mu_{J}(F)$$\mu_{A}(F)$$\beta_{A}(F)$$\mu_{J}(F)$$\mu_{A}(F)$and $\mu_{A}’(F)\leq 0$ . Then equation (2.2) has exactly one root whenever the left

hand side exceeds 1 for large $F$ .

37
2.2

Linearized equations

Throughout the remainder of this paper, we as sume that the interior equi-

librium uniquely exists. In this subsection, we derive linearized equations for
system (DE) around the interior equilibrium$\overline{F}$and . We do not write calcu$\overline{b}$lations for deriving the linearized equations. We shall only show the results.

Define$\theta_{J}(\overline{F}):=\overline{b}\mu_{J}(\overline{F})\{\frac{g’(\overline{F})}{g(\overline{F})}-\frac{\mu_{J}’(\overline{F})}{\mu_{J}(\overline{F})}\}$$k_{10}= \overline{b}\frac{\beta_{A}’(\overline{F})}{\beta_{A}(\overline{F})},$,$\theta_{A}(\overline{F}):=\overline{b}\mu_{A}(\overline{F})\{\frac{g’(\overline{F})}{g(\overline{F})}-\frac{\mu_{A}’(\overline{F})}{\mu_{A}(\overline{F})}\}$.

,$k_{11}(\sigma)=H(\sigma-\overline{\tau})\mu_{A}(\overline{F})e^{\mu_{A}(F)(\overline{\tau}-\sigma)}$where $H$ is the Heaviside function,$k_{12}(\sigma)=\{\begin{array}{ll}\theta_{J}(\overline{F})+(\theta_{A}(\overline{F})-\theta_{J}(\overline{F}))e^{-\mu A(\overline{F})\sigma} for \sigma\leq\overline{\tau}\theta_{J}(\overline{F})-\overline{b}\mu_{A}(\overline{F})\frac{g’(\overline{F})}{g(\overline{F})}+(\theta_{A}(\overline{F})- (\overline{F}))e^{-\mu_{A}(\overline{F})\sigma} for \sigma>\overline{\tau},\end{array}$$k_{20}=- \overline{b}\{\frac{\gamma_{A}’(\overline{F})}{\mu_{A}(\overline{F})}e^{-\mu_{J}(\overline{F})\overline{\tau}}+\frac{\gamma_{J}’(\overline{F})}{\mu_{J}(\overline{F})}(1-e^{-\mu_{J}(\overline{F})\overline{\tau}})\}$,$k_{21}(\sigma)=\{\begin{array}{ll}-\gamma_{J}(\overline{F})e^{-\mu_{J}(\overline{F})\sigma} for \sigma\leq\overline{\tau}-\gamma_{A}(\overline{F})e^{-\mu_{J}(\overline{F})\overline{\tau}}e for \sigma>\overline{\tau}\end{array}$,$k_{22}( \sigma)=-\frac{\gamma_{A}(\overline{F})}{\beta_{A}(\overline{F})}k_{12}(\sigma)+\gamma_{J}(\overline{F})\overline{b}e^{-\mu_{J}(\overline{F})\overline{\tau}}\varphi(\sigma)$$\varphi(\sigma)=\{\begin{array}{ll}\frac{g’(\overline{F})}{g(\overline{F})}-\frac{\mu_{J}’(\overline{F})}{\mu_{J}(\overline{F})}\{1-e^{-\mu_{J}(\overline{F})(\sigma-\overline{\tau})}\} for \sigma\leq\overline{\tau}0 for \sigma>\overline{\tau}.

\end{array}$

The linearized system is given by$x(t)=k_{10}y(t)+ \int_{0}^{\infty}(k_{11}(\sigma)x(t-\sigma)+k_{12}(\sigma)y(t-\sigma))d\sigma$

,

(2.3)

$\frac{dy}{dt}(t)=k_{20}y(t)+\int_{0}^{\infty}(k_{21}(\sigma)x(t-\sigma)+k_{22}(\sigma)y(t-\sigma))d\sigma$

.

(2.4)

38
Characteristic equation

2.3

The characteristic equation for (2.3) and (2.4) is given by

(2.5)

,

$\{1-\tilde{k}_{11}(\lambda)\}\{\lambda-k_{20}-\tilde{k}_{22}(\lambda)\}-\tilde{k}_{21}(\lambda)\{k_{10}+\tilde{k}_{12}(\lambda)\}=0$

where$\tilde{k}_{mn}(\lambda)=\int_{0}^{\infty}k_{mn}(\sigma)e^{-\lambda\sigma}d\sigma$,$(m, n=1,2)$ .

(2.6)

To check whether equations (DE) undergoes a Hopf bifurcation, we look for acomplex conjugate of pure imaginary root of (2.5). System (DE) defines non-sun-reflexive dual semigroups in a non-reflexive Banach space, so the theory
on sun-reflexive dual semigroups has supposed not to work if one only refers
the book for delay equations [2]. However, recently Diekmann and Gyllenberg have shown that delay equations can be reformulated as abstract weak-’-integral equations involving dual semigroups when the solution take values

in a non-reflexive Banach space [4]. Then the theory, methods and results

such as linearized stability, center manifold theory and Hopf bifurcation theory developed in [2] are applicable to our delay equations (see also [2], [3], [4]).

Hereafter we suppress to write the argument and simply write,and . For$\mu_{J}$$\beta_{A},$$\gamma_{A},$$\gamma_{J},$$g$,$\omega\neq 0$$\mu_{A}$$\tilde{k}_{mn}(i\omega)=c_{mn}(\omega)-is_{mn}(\omega)$$:= \int_{0}^{\infty}k_{mn}(\sigma)$

Define$\varphi_{A}(\omega)\in(0, \pi/2)$cos$( \omega\sigma)d\sigma-i\int_{0}^{\infty}k_{mn}(\sigma)$

and$\varphi_{J}(\omega)\in(0, \pi/2)$

and

sin

$(\omega\sigma)d\sigma$$\sin\varphi_{A}(\omega)=\frac{\omega}{\sqrt{\mu_{A}^{2}+\omega^{2}}}$

(2.7)

.

(2.8)

$\varphi_{A}(\omega)=\frac{\mu_{A}}{\sqrt{\mu_{A}^{2}+\omega^{2}}}$COS$\varphi_{J}(\omega)=\frac{\mu_{J}}{\sqrt{\mu_{J}^{2}+\omega^{2}}}nnd\sin\varphi_{J}(\omega)=\frac{\omega}{\sqrt{\mu_{J}^{2}+\omega^{2}}}$$\varphi_{A}(\omega)$and$\varphi_{J}(\omega)$as$\frac{\cos\varphi_{A}}{\mu_{A}}=\frac{\sin\varphi_{A}}{\omega}$$\varphi_{A}$

and

and

$\varphi_{J}$, respectively. Note that$\frac{\cos\varphi_{J}}{\mu_{J}}=\frac{\sin\varphi_{J}}{\omega}$

Direct calculation yields thatcos$(\omega\tau+\varphi_{A}),$$s_{11}=\cos\varphi_{A}$$(m, n=1,2)$ .

by

COS

We simply write$c_{11}=\cos\varphi_{A}$,sin$(\omega\tau+\varphi_{A})$

,

.

(2.9)

39
$c_{12}= \theta_{J^{\frac{\sin\omega\tau}{\omega}+\frac{\sin\varphi_{A}}{\omega}}}[(\theta_{A}-\theta_{J})\cos\varphi_{A}+(\theta_{J}-b\mu_{A}\frac{g’}{g})\cos(\omega\tau+\varphi_{A})]$,$s_{12}= \theta_{\frac{1-\cos\omega\tau}{\omega}+\frac{\sin\varphi_{A}}{\omega}}[(\theta_{A}-\theta_{J})\sin\varphi_{A}+(\theta_{J}-b\mu_{A}\frac{g’}{g})\sin(\omega\tau+\varphi_{A})]$,,$c_{21}=- \frac{\gamma_{A}}{\mu_{A}}e^{-\mu_{J^{\mathcal{T}}}}\cos\varphi_{A}\cos(\omega\tau+\varphi_{A})-\frac{\gamma_{J}}{\mu_{J}}\cos\varphi_{J}[\cos\varphi_{J}-e^{-\mu_{J^{\mathcal{T}}}}\cos(\omega\tau+\varphi_{J})]$$s_{21}=- \frac{\gamma_{A}}{\mu_{A}}e^{-\mu_{J^{\mathcal{T}}}}\cos\varphi_{A}\sin(\omega\tau+\varphi_{A})-\frac{\gamma_{J}}{\mu_{J}}\cos\varphi_{J}[\sin\varphi_{J}-e^{-\mu_{J^{\mathcal{T}}}}\sin(\omega\tau+\varphi_{J})]$,$c_{22}=( \frac{\gamma_{J}}{\mu_{J}}-\frac{\gamma_{A}}{\mu_{A}})e^{-\mu_{J^{\mathcal{T}}}}\theta_{J}\frac{\sin\omega\tau}{\omega}+b\frac{\mu_{J}’}{\mu_{J}}\frac{\gamma_{J}}{\mu_{J}}\cos\varphi_{J}[\cos\varphi_{J}-e^{-\mu_{J}\tau}\cos(\omega\tau+\varphi_{J})]$,$- \frac{\gamma_{A}}{\mu_{A}}e^{-\mu_{J^{\mathcal{T}_{\frac{\sin\varphi_{A}}{\omega}}}}}[(\theta_{A}-\theta_{J})\cos\varphi_{A}+(\theta_{J}-b\mu_{A}\frac{g’}{g})\cos(\omega\tau+\varphi_{A})]$$s_{22}=( \frac{\gamma_{J}}{\mu_{J}}-\frac{\gamma_{A}}{\mu_{A}})e^{-\mu_{J^{\mathcal{T}}}}\theta_{\sqrt{},\omega}+b\frac{\mu_{J}’}{\mu_{J}}\frac{\gamma_{J}}{\mu_{J}}\cos\varphi_{J}[\sin\varphi_{J}-e^{-\mu_{J^{\mathcal{T}}}}\sin(\omega\tau+\varphi_{J})]1-\cos\omega\tau$$- \frac{\gamma_{A}}{\mu_{A}}e^{-\mu_{J^{\mathcal{T}_{\frac{\sin\varphi_{A}}{\omega}}}}}[(\theta_{A}-\theta_{J})\sin\varphi_{A}+(\theta_{J}-b\mu_{A}\frac{g’}{g})\sin(\omega\tau+\varphi_{A})]$.

Here we exploited relations$\int_{0}^{\tau}e^{-\mu\sigma}\cos\omega\sigma d\sigma=\frac{\mu-e^{-\mu\tau}(\mu\cos\omega\tau-\omega\sin\omega\tau)}{\mu^{2}+\omega^{2}}$,$\int_{0}^{\tau}e^{-\mu\sigma}\sin\omega\sigma d\sigma=\frac{\omega-e^{-\mu\tau}(\omega\cos\omega\tau+\mu\sin\omega\tau)}{\mu^{2}+\omega^{2}}$,$l^{\infty}e^{-\mu\sigma} \cos\omega\sigma d\sigma=\frac{e^{-\mu\tau}(\mu\cos\omega\tau-\omega\sin\omega\tau)}{\mu^{2}+\omega^{2}}$,$l^{\infty}e^{-\mu\sigma} \sin\omega\sigma d\sigma=\frac{e^{-\mu\tau}(\omega\cos\omega\tau+\mu\sin\omega\tau)}{\mu^{2}+\omega^{2}}$and$e^{-\mu_{J}\tau}=\mu_{A}/\beta_{A}$. Introducing complex variables$z_{mn}$$:=c_{mn}+is_{mn},$(2.10)$(m, n=1,2)$

gives$z_{11}=\cos\varphi_{A}e^{i(w\tau+\varphi_{A})}$,$z_{12}= \frac{\theta_{J}}{\omega}2sn(\frac{\omega\tau}{2})e^{i^{w_{2}}}\pm+\frac{\sin\varphi_{A}}{\omega}e^{i\varphi_{A}}\{(\theta_{A}-\theta_{J})+(\theta_{J}-b\mu_{A}\frac{g’}{g})e^{iw\tau}\}$$z_{21}=- \frac{\gamma_{A}}{\mu_{A}}e^{-\mu_{J^{\mathcal{T}}}}$

cos$\varphi_{A}e^{i(\omega\tau+\varphi_{A})}-\frac{\gamma_{J}}{\mu_{J}}\cos\varphi_{J}e^{\mathfrak{i}\varphi_{J}}(1-e^{-\mu_{J}\tau}e^{iw\tau})$$z_{22}= \frac{\theta_{J}}{\omega}(\frac{\gamma_{J}}{\mu_{J}}-\frac{\gamma_{A}}{\mu_{A}})e^{-\mu_{J}\tau}2$sin$( \frac{\omega\tau}{2})e^{i\frac{w\tau}{2}}+b\frac{\mu_{J}’}{\mu_{J}}\frac{\gamma_{J}}{\mu_{J}}$

cos

,

,

$\varphi_{J}e^{i\varphi_{J}}(1-e^{-\mu_{J}\tau}e^{\dot{u}d\tau})$$- \frac{\gamma_{A}}{\mu_{A}}e^{-\mu_{J^{\mathcal{T}}}}\frac{\sin\varphi_{A}}{\omega}e^{i\varphi_{A}}\{(\theta_{A}-\theta_{J})+(\theta_{J}-b\mu_{A}\frac{g’}{g})e^{iw\tau}\}$.

40
Note that$z_{12}z_{21}-z_{11}z_{22}=\{-(c_{11}c_{22}-s_{11}s_{22})+c_{12}c_{21}-s_{12}s_{21}\}$$+i\{-(s_{11}c_{22}+c_{11}s_{22})+(s_{12}c_{21}+c_{12}s_{21})\}$

Therefore, characteristic equation (2.5) with$\lambda=i\omega$can be rewritten as$k_{20}+i\omega-(k_{20}+i\omega)z_{11}+k_{10}z_{21}+z_{22}+z_{12}z_{21}-z_{11}z_{22}=0$

2.4

.

.

(2.11)

Food dependent uptake ratedepend on theandIn this subsection, we suppose that only uptake rates
food density. Then it immediately foUows that $\mu_{J}’=\mu_{A}’=\beta_{A}’=g’=0$ . Note$\gamma_{A}$$\gamma_{J}$that $k_{10}=\theta_{A}=\theta_{J}=0$ .

$z_{11},$$z_{11}=\cos\varphi_{A}e^{i(\omega\tau+\varphi_{A})},$$z_{21}=- \frac{\gamma_{A}}{\mu_{A}}e^{-\mu_{J}\tau}$cos$z_{12},$$z_{21}$and$z_{22}$$z_{12}=z_{22}=0$

are reduced to,$\varphi_{A}e^{i(\omega\tau+\varphi A)}-\frac{\gamma_{J}}{\mu_{J}}\cos\varphi_{J}e^{i\varphi_{J}}(1-e^{-\mu_{J^{\mathcal{T}}}}e^{iw\tau})$.

Thus $z_{12}z_{21}-z_{11}z_{22}=0$ . $(2.11)$ is reduced to$k_{20}+i\omega-(k_{20}+i\omega)z_{11}=0$

Substituting$z_{11}$and$z_{21}$$e^{\dot{u}d\mathcal{T}}|=1$(2.12)

into (2.12) gives$(k_{20}+i\omega)$

Since 1.

cos$\varphi_{A}e^{i\varphi_{A}}e^{iw\tau}=k_{20}+i\omega$,(2.13),

1$(k_{20}+i\omega)$cos$\varphi_{A}e^{i\varphi_{A}}|=|k_{20}+i\omega|$.

(2.14)

It follows from (2.14) that

1 cos$\varphi_{A}e^{i\varphi_{A}}|=\frac{\mu_{A}}{\mu_{A}^{2}+\omega^{2}}=1$.

. This is a contradiction. Hence there are no
Now we are assuming thatpure imaginary roots of the characteristic equation.
$\omega\neq 0$

41Discussion

3

We studied delay equations with infinite delay describing the population dynamics of size-structured population which is feeding on some food as a re-source. We showed the condition for the existence of the interior equilibriumand derived the linearized equations for system (DE) around the interior equilibrium. The characteristic equation is defined for the linearized system. We

focused on a particular case to investigate the occurrence of population cycle
via Hopf bifurcation. In a particular case, the rates for growth , reproduction
are independent functions ofand death of adults, death of juveniles
the food density, while consumption rates for juveniles and adults are func$g$$\beta_{A}$$\mu_{J}$$\mu_{A}$tions of the food density. In this case, we showed that no occurrence of Hopf
bifurcation is expected. This finding suggests that for the occurrence of Hopf
bifurcation, we should lmpose that at least one physiological rates
depends on the food density $F$ . If we do not assume thator$\beta_{A},$$g,$$\mu_{J}$$\beta_{A},$$g,$$\mu_{J}$$\mu_{A}$are independent functions of the food density, we may expect there
exists a pair of complex conjugate of pure imaginary roots of characteristic
equation. Actually, extensive numerical computations and numerical simulaand$\mu_{A}$tions implemented in [6] showed the occurrence of sustained population cycles.
Mathematical analysis to show the existence of pure imaginary roots of thecharacteristic equation for general case is left for our future work.

References[1] Cooke, K.L., van den Driessche, P., Zou, X., Interaction of maturation
delay and nonlinear birth in population and epidemic models. J. Math.

Biol. 39, 332-352 (1999)[2] Diekmann $0.$ , van Gilis, S.A., Lunel, S.M.V., Walther, H.-O., Delay Equations: Functional, Complex, and Nonlinear Analysis, volume 110 of Applied Mathematical Sciences. Springer-Verlag, 1995.

42[3] Diekmann, O., Getto, Ph., Gyllenberg, M., Stability and bifvrcation analysis of Volterra functional equations in the light of suns and stars, submitted.

[4] Diekmann, Gyllenberg, M., Abstract delay equations inspired by popula-tion dynamics, submitted.
[5] Gourley, S.A. and Kuang,$Y$, A stage structured predator-prey model andits dependence on maturation delay and death rate. J. Math. Biol. 49,

188-200 (2004)[6] de Roos, A.M. and Persson, L., Competition in size-structured popu-

lations: mechanisms inducing cohort formation and population cycles.

Theor. Pop. Biol. 63, 1-16 (2003)[7] Werner, E. E., Size, scaling and the evolution of complex life cycles. In
Ebenman, B., and Persson, L., editors, Size-stretctured Populations-Ecol-ogy and Evolution, pp 60-81, Springer Verlag, Berlin. 1988.