J. Math. Biol. 49, 188–200 (2004)
Digital Object Identifier (DOI):
10.1007/s00285-004-0278-2
Mathematical Biology
Stephen A. Gourley · Yang Kuang
A stage structured predator-prey model and its
dependence on maturation delay and death rate
Received: 24 January 2003 / Revised version: 26 February 2004 /
c Springer-Verlag 2004
Published online: 31 May 2004 – Abstract. Many of the existing models on stage structured populations are single species
models or models which assume a constant resource supply. In reality, growth is a combined
result of birth and death processes, both of which are closely linked to the resource supply
which is dynamic in nature. From this basic standpoint, we formulate a general and robust
predator-prey model with stage structure with constant maturation time delay (through-stage
time delay) and perform a systematic mathematical and computational study. Our work indicates that if the juvenile death rate (through-stage death rate) is nonzero, then for small and
large values of maturation time delays, the population dynamics takes the simple form of
a globally attractive steady state. Our linear stability work shows that if the resource is
dynamic, as in nature, there is a window in maturation time delay parameter that generates
sustainable oscillatory dynamics.
1. Introduction and preliminaries
Many consumer species go through two or more life stages as they proceed from
birth to death. The majority of the models in the literature ignore such reality and
lump individuals into one single reproducing category which can be modeled by a
single ordinary differential equation (ODE). Unfortunately, such simple ODEs are
only capable of generating simple equilibrium dynamics. In order to capture the
oscillatory behavior often observed in nature, various mechanisms are proposed.
Such mechanisms include difference models (May [16]) and delay differential models (Murdoch et al [17], Kuang [14]). While difference models are suitable for
non-overlapping generations, such models are forced to ignore dynamics between
generations (for such populations, birth may take place in a short time frame, but
death occurs continuously) and tend to ignore the dynamics of resources. On the
other hand, the discrete delay logistic equation (the so-called Hutchinson equation) - the traditional prototype for single species continuous growth model - is ill
S.A. Gourley: Department of Mathematics and Statistics, University of Surrey, Guildford,
Surrey GU2 7XH, UK. e-mail: s.gourley@surrey.ac.uk
Y. Kuang: Department of Mathematics and Statistics, Arizona State University, Tempe, AZ
85287, USA. e-mail: kuang@asu.edu.
Work is partially supported by NSF grant DMS-0077790.
Mathamatics Subject Classification (2000): 92D25, 35R10
Key words or phrases: Delay equation – Stage structure – Intraspecific competition – Lyapunov functional – Population model – Through-stage death rate
A Stage structured predator-prey dynamics
189
formulated and produces artificially complex dynamics such as excessive volatility
and huge peak to valley ratios (p97 May [16]; see also Gourley and Kuang [7]).
For example, when rT = 2.5, the peak to valley ratio of the Hutchinson equation
N (t) = rN (t)[1 − N (t − T )/K] is 2930.
Of special relevance to the present paper is the work in the 1980’s of Gurney,
Nisbet and others who proposed and analysed various models known as ‘stagestructured’ models. See, for example, Gurney and Nisbet [8], Gurney et al [9],
Nisbet and Gurney [18] and Nisbet et al [19, 20]. These authors were particularly
concerned with the need for a systematic approach to model formulation, and with
the need for models containing parameters measured by ecologists. For example,
in [8] a larval maturation time model is proposed and analysed, and it was shown that
time delays can destabilise the system leading to limit cycles containing multiple
overlapping generations. Later, in [12], estimates were obtained for the period to
delay ratio near stability switches. A good overview on stage-structured models can
be found in the recent book by Murdoch et al [17] (Chpt. 5 in particular).
The work of Aiello and Freedman [1] on a single species stage-structured model
has received much attention in the more mathematically oriented literature. Like
the work of Gurney and Nisbet and their co-workers, it aims for a mathematically
careful and ecologically meaningful model formulation approach. Their model
predicts a positive steady state as the global attractor, thereby suggesting that stage
structure does not generate the sustained oscillations frequently observed in nature
in single populations. This work inevitably stirred some controversy. Subsequent
works by other authors (e.g. Aiello et al [2], Arino et al [3]) suggest that the time
delay to adulthood should be state dependent and careful formulation of such state
dependent time delays can lead to models that produce periodic solutions (Arino
et al. [3]).
We review this work from a different yet simple and basic biological angle:
consumer species growth is a combined result of birth and death processes, both of
which are closely linked to the resource supply, and resource supply is renewable
and is dynamically linked to the consumer growth dynamics. From this point of
view, we formulate a general and robust stage structured predator-prey model with
constant maturation time delay and perform a systematic mathematical study. Our
work indicates that for small and large values of the maturation time delays, the
population dynamics takes the simple form of a globally attractive steady state.
However, our linear stability work shows that if the resource is dynamic, as in
nature, there is a window in maturation time delay parameter that generates sustainable oscillatory dynamics. Qualitatively similar findings were reported by Cooke
et al. [5] in their study of a scalar delay equation with delay in the (nonlinear) births
term.
To facilitate the interpretation of our mathematical findings and possible lab
or field implementation of the model, we assume that the prey or the renewable
resource, denoted by x, can be modeled by a logistic equation when the consumer
is absent. As in the work of Aiello and Freedman [1], we assume that the predators
or consumers are divided into two stage groups: juveniles and adults and they are
denoted by yj and y respectively below. As in Aiello and Freedman [1], we assume
that only adult predators are capable of preying on the prey species and that the
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S.A. Gourley, Y. Kuang
juvenile predators live on their parents, or on a resource that is different from that
required by adults and available in abundance. With these assumptions, we have
the following plausible two-stage predator-prey (resource-consumer) interaction
model:
x (t) = rx(t)(1 − x(t)/K) − y(t)p(x(t)),
y (t) = b e−dj τ y(t − τ )p(x(t − τ )) − dy(t),
yj (t) = by(t)p(x(t)) − b e−dj τ y(t − τ )p(x(t − τ )) − dj yj (t),
x(θ ), y(θ ) ≥ 0 are continuous on − τ ≤ θ < 0, and x(0), y(0), yj (0) > 0.
(1.1)
Here r is the specific growth rate of the prey and K is its carrying capacity. The
function p(x) is the adult predators’ functional response. The parameters b and d
are the adult predators’ birth and death rates, respectively. In addition, we assume
that juveniles suffer a mortality rate of dj (the through-stage death rate) and take
τ units of time to mature. Notice that, mathematically, no information on the past
history of yj is needed for system (1.1).
The dynamics of model (1.1) are determined by the first two equations. Therefore, in the rest of this paper, we will study the following
x (t) = rx(t)(1 − x(t)/K) − y(t)p(x(t)),
y (t) = b e−dj τ y(t − τ )p(x(t − τ )) − dy(t),
x(θ ), y(θ ) ≥ 0 are continuous on − τ ≤ θ < 0, and x(0), y(0) > 0.
(1.2)
Now, both r and K of system (1.2) can be easily scaled off by appropriate rescaling
of time and the x variable. Therefore, in the following, we will consider the system
x (t) = x(t)(1 − x(t)) − y(t)p(x(t)),
y (t) = b e−dj τ y(t − τ )p(x(t − τ )) − dy(t),
(1.3)
where b, dj , d > 0 and τ ≥ 0. The function p(x) is assumed to be differentiable
and satisfy
p(0) = 0,
p(x) is strictly increasing,
p(x)/x bounded for all x ≥ 0.
(1.4)
Reasonable choices for p(x) include the cases p(x) = px (p > 0 constant), and
p(x) = px/(1 + ax) (p, a > 0). Let us prove the following positivity preservation
result.
Proposition 1. Let x(θ ), y(θ ) ≥ 0 on −τ ≤ θ < 0, and x(0), y(0) > 0. Then the
solution of (1.3), with p(x) satisfying (1.4), satisfies x(t), y(t) > 0 for all t > 0.
Proof. The assumptions on p(x) imply that the right hand side of the equation
for x effectively contains a factor of x(t). Positivity for x(t) therefore follows by
standard arguments. For y(t) note that, on 0 ≤ t ≤ τ ,
y (t) ≥ −dy(t).
Hence, y(t) > y(0)e−dt > 0 for all t ∈ [0, ∞). The proof of the proposition is
complete.
Remark. Of course, positivity of x(t) and y(t) in system (1.1) now follows immediately.
A Stage structured predator-prey dynamics
191
y
x−isocline
the y−isocline,
as τ is increased
(x*,y*)
1
0
x
Fig. 1. The x and y-isoclines, for various values of τ , of the system obtained from (1.3) by
removing the delays from the arguments
2. Equilibria and their feasibility
Apart from the zero solution, system (1.3) always has (x, y) = (1, 0) as an
equilibrium. The components of any interior equilibrium must satisfy
y=
x(1 − x)
,
p(x)
b e−dj τ p(x) = d.
(2.1)
Whether an interior equilibrium (x ∗ , y ∗ ) is feasible or not depends on the values
of the parameters. Fig. 1 shows the x- and y- isoclines of the system obtained
from (1.3) by removing the delays from the arguments. It is clear that an interior
equilibrium will exist if and only if
b e−dj τ p(1) > d.
(2.2)
Fig. 1 also makes clear how the interior equilibrium (x ∗ , y ∗ ), if feasible, depends
on the delay τ and the other parameters. Note, for example, that condition (2.2)
can only possibly be satisfied for τ up to a certain finite value. Increasing τ lowers
the y-isocline in Fig. 1, causing the coincidence of (x ∗ , y ∗ ) with (1, 0) at a finite
value of τ . For higher τ there is no interior equilibrium. In all our analysis it will be
important to keep track of how the interior equilibrium depends on the parameters.
3. Global stability of the equilibrium (1, 0)
We will need the following simple and well known result, which is a direct application of Theorem 4.9.1 in Kuang ([14], p159)
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S.A. Gourley, Y. Kuang
Lemma 1. If a < b, then the solution of the equation
u (t) = au(t − τ ) − bu(t),
(3.1)
where a, b, τ > 0, and u(t) > 0 for −τ ≤ t ≤ 0, satisfies limt→∞ u(t) = 0.
The next result gives conditions which are both necessary and sufficient for
the global stability of the boundary equilibrium (x, y) = (1, 0). The biological
meaning of the condition is obvious: if the adult predators’ recruitment rate at the
peak of prey abundance is no more than their death rate, then the predators face
extinction.
Theorem 1. Assume that be−dj τ p(1) ≤ d. Then solutions of (1.3) satisfy x(t) → 1,
y(t) → 0 as t → ∞.
Proof. We consider first the case when be−dj τ p(1) < d. Then there exists ε > 0
such that be−dj τ p(1 + ε) < d. Also, by positivity of solutions, x (t) ≤ x(t)(1 −
x(t)). This implies that lim supt→∞ x(t) ≤ 1 and therefore there exists Tε > 0
such that x(t) < 1 + ε for all t ≥ Tε . Then, for t ≥ Tε + τ ,
dy(t)
= b e−dj τ y(t − τ )p(x(t − τ )) − dy(t)
dt
≤ b e−dj τ y(t − τ )p(1 + ε) − dy(t)
By comparison, y(t) is bounded above by the solution u(t) of
u (t) = b e−dj τ u(t − τ )p(1 + ε) − du(t),
t > Tε + τ
satisfying u(t) = y(t) for t ∈ [Tε , Tε + τ ]. Since be−dj τ p(1 + ε) < d, Lemma 1
yields that u(t) → 0. Hence y(t) → 0.
Let η ∈ (0, 1). Then there exists Tη > 0 such that, for t ≥ Tη ,
y(t) p(x(t)) = y(t)x(t)
p(x(t))
< ηx(t)
x(t)
by boundedness of p(x)/x. Then, for t ≥ Tη ,
dx(t)
≥ x(t)(1 − η − x(t)).
dt
By another comparison argument,
lim inf x(t) ≥ 1 − η.
t→∞
Since η ∈ (0, 1) was arbitrary, lim inf t→∞ x(t) ≥ 1. We already have
lim supt→∞ x(t) ≤ 1. Hence limt→∞ x(t) = 1.
Next, we shall consider the case when be−dj τ p(1) = d. Since dx(t)/dt ≤
x(t)(1 − x(t)), x(t) is always decreasing when above 1. If x(t) should ever get
below 1 then x(t) must stay strictly below 1 for all subsequent time (x(t) cannot
even ‘touch’ 1 again since we’d have to have x = 0 at such a time and the differential equation for x makes clear the impossibility of this). This implies there are two
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possible scenarios, either (i) x(t) → 1 from above as t → ∞, or (ii) there exists T
such that x(t) < 1 for all t > T .
If the first of these scenarios applies then we have only to show that y(t) → 0.
Integrating the equation for x in (1.3),
t
t
x(s)(1 − x(s)) ds −
y(s) p(x(s)) ds
x(t) − x(0) =
0
0
t
≤
≥1
t
x(s)(1 − x(s)) ds − p(1)
0
y(s) ds.
0
Therefore,
p(1)
t
y(s) ds ≤
0
0
t
x(s)(1 − x(s)) ds + x(0) − x(t) ≤ x(0).
≤0
Letting t → ∞, we conclude that y(s) ∈ L1 (0, ∞) and, therefore, y(t) → 0.
In the second scenario described above, x(t) < 1 for all t > T and therefore,
for t > T + τ ,
dy(t)
= b e−dj τ y(t − τ )(p(x(t − τ )) − p(1)) + b e−dj τ y(t − τ )p(1) − dy(t)
dt
= b e−dj τ y(t − τ )(p(x(t − τ )) − p(1)) + dy(t − τ ) − dy(t).
Consider the functional
V = y (t) + d
2
t
y 2 (s) ds.
t−τ
Then, for t sufficiently large,
dV
= 2y(t)b e−dj τ y(t − τ )(p(x(t − τ )) − p(1))
dt
+2dy(t)y(t − τ ) − dy 2 (t) − dy 2 (t − τ )
= 2y(t)b e−dj τ y(t − τ )(p(x(t − τ )) − p(1)) − d(y(t − τ ) − y(t))2 < 0.
A direct application of the well known Liapunov-LaSalle type theorem (Theorem
2.5.3 of Kuang [14], p30) shows that limt→∞ y(t) = 0 and limt→∞ x(t) = 1. The
proof of the theorem is complete.
4. Linearised analysis
We shall carry out the analysis for general p(x) as far as possible, but will concentrate on the case p(x) = px later to enable further analytic progress and for
the purposes of numerical simulation. In particular, we will compare and contrast
the dynamics in the two cases when dj > 0 and dj = 0. These cases correspond
respectively to the presence or absence of mortality among the juveniles.
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We will begin by examining the linear stability of the equilibrium (x ∗ , y ∗ ),
assuming of course that
be−dj τ p(1) > d
so that the equilibrium is feasible. If dj > 0 (i.e. there is mortality among the
juveniles) then, as one increases the delay τ , the equilibrium loses feasibility at a
finite value of τ . If dj = 0 the equilibrium is either feasible for all τ , or not feasible
at all.
Let us linearise (1.3) at the interior equilibrium (x ∗ , y ∗ ). Setting x = x ∗ + u,
y = y ∗ + v where u and v are small, and linearising, gives
u (t) = (1 − 2x ∗ − y ∗ p (x ∗ ))u(t) − p(x ∗ )v(t),
v (t) = be−dj τ y ∗ p (x ∗ )u(t − τ ) + be−dj τ p(x ∗ )v(t − τ ) − dv(t).
(4.1)
Non-trivial solutions of the form (u, v) = (c1 , c2 ) exp(λt) exist if and only if
G(λ, τ ) = 0, where
G(λ, τ ) = λ2 + (2x ∗ + y ∗ p (x ∗ ) − 1 + d − be−dj τ p(x ∗ )e−λτ )λ
+(2x ∗ + y ∗ p (x ∗ ) − 1)(d − be−dj τ p(x ∗ )e−λτ )
+be−dj τ y ∗ p(x ∗ )p (x ∗ )e−λτ .
(4.2)
For now, our interest will be in the possibility of stability switches as τ is increased
from zero, and therefore we shall assume that the interior equilibrium is stable when
τ = 0. When τ = 0, the characteristic equation simplifies to
λ2 + (2x ∗ + y ∗ p (x ∗ ) − 1)λ + by ∗ p(x ∗ )p (x ∗ ) = 0
(we have used the fact that when τ = 0 the equilibrium’s components are related
by bp(x ∗ ) = d). For stability, we require the coefficient of λ and the constant
term to be strictly positive (note in particular that this is assured if we choose the
parameters so that x ∗ > 1/2).
Let us turn to the case τ > 0. Examining the characteristic equation G(λ, τ ) =
0, one notices that τ does not appear only in the e−λτ terms, but also in numerous
other places, since x ∗ and y ∗ depend on τ . This is because our model equations (1.3)
have delay dependent parameters. Delay systems in which the coefficients are independent of the time delays are simpler to study; in such cases the characteristic
equation involves the delay only through the e−λτ term. The theory in such cases is
very well developed (see, for example, Kuang [14]) and one can usually compute
exactly the values of τ at which stability switches occur. In our situation there are
two main complications. The first of these is that when one searches for purely
imaginary roots λ = ±iω of the characteristic equation, the polynomial equation
that one obtains for ω still has τ in its coefficients (this is not so in the welldeveloped theory for systems with delay-independent parameters). This effectively
means that one cannot compute exactly the values of τ at which stability switches
occur. The second serious complication is that even the equilibrium components x ∗
and y ∗ involve τ . It is important to know the dependence of these components on τ ,
yet often the components cannot be computed explicitly (though we can, and will,
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compute x ∗ and y ∗ explicitly in the case when p(x) = px). Also, when looking for
stability switches, the equilibrium’s feasibility, which is again τ -dependent, must
be kept track of.
Since delay equations with delay-dependent parameters are now arising somewhat frequently in the literature, Beretta and Kuang [4] have developed a systematic
approach to studying the difficult characteristic equations arising from such systems. Their approach is a computationally assisted one, requiring the plotting of
accurate graphs of certain functions. One cannot in practice compute the stability switches analytically. We shall summarize their technique as it applies to our
particular problem.
The eigenvalue equation G(λ, τ ) = 0 can be written as
P (λ, τ ) + Q(λ, τ )e−λτ = 0
where
P (λ, τ ) = λ2 + (2x ∗ + y ∗ p (x ∗ ) − 1 + d)λ + d(2x ∗ + y ∗ p (x ∗ ) − 1),
Q(λ, τ ) = −be−dj τ p(x ∗ )λ − be−dj τ p(x ∗ )(2x ∗ − 1).
We seek purely imaginary roots λ = iω of G(λ, τ ) = 0, where ω = min{a :
a > 0, G(ia, τ ) = 0}. Clearly, ω is unique if exists. We have P (iω, τ ) =
−Q(iω, τ )e−iωτ . Taking the complex conjugate of this, and then eliminating e−iωτ ,
yields
|P (iω, τ )|2 = |Q(iω, τ )|2
which determines ω in terms of τ , ω = ω(τ ). Also
PR (iω, τ ) + QR (iω, τ ) cos ωτ + QI (iω, τ ) sin ωτ = 0
and
PI (iω, τ ) + QI (iω, τ ) cos ωτ − QR (iω, τ ) sin ωτ = 0
where the subscripts denote real and imaginary parts. Hence
sin(ω(τ )τ ) =
PI (iω(τ ), τ )QR (iω(τ ), τ ) − PR (iω(τ ), τ )QI (iω(τ ), τ )
(4.3)
Q2R (iω(τ ), τ ) + Q2I (iω(τ ), τ )
and
cos(ω(τ )τ ) = −
PR (iω(τ ), τ )QR (iω(τ ), τ ) + PI (iω(τ ), τ )QI (iω(τ ), τ )
.
Q2R (iω(τ ), τ ) + Q2I (iω(τ ), τ )
(4.4)
Define the function θ (τ ) ∈ [0, 2π) such that sin θ (τ ) and cos θ(τ ) are given by
the right-hand sides of (4.3) and (4.4) respectively. Then the τ we seek, at which
stability switches occur, are the solutions of
Sn (τ ) := τ − τn (τ ) = 0
(4.5)
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S.A. Gourley, Y. Kuang
2
Graph of stability switch for S_0(T) and S_1(T), here T=tau
0.5
T
1.5
1
2
2.5
3
0
–2
–4
–6
y
–8
–10
–12
–14
Fig. 2. Plots of the functions S0 (τ ) (upper curve) and S1 (τ ) (lower curve), when p(x) = px.
Parameter values used are p = 1, b = 10, dj = 1 and d = 0.5. The equilibrium is feasible
for 0 ≤ τ < (1/dj ) ln(bp/d) ≡ τe ≈ 3. The first vertical line provides the end point for the
existence interval for S0 (τ ) and the second one is the line τ = τe .
τ=0.1
τ=0.6
3.5
5
prey
predator
3
prey
predator
4
3
2
x, y
x, y
2.5
1.5
2
1
1
0.5
0
0
100
200
0
300
0
100
τ=1.2
200
300
τ=1.8
3
2
prey
predator
2.5
prey
predator
1.5
x, y
x, y
2
1.5
1
1
0.5
0.5
0
0
100
200
time t
300
0
0
100
200
300
time t
Fig. 3. A solution of model (1.3) with p(x) = px, x(θ) = 0.3, y(θ) = 1, θ ∈ [−τ, 0]
where p = 1.0, b = 10, dj = 1.0, d = 0.5 and τ varies from 0.4 to 1.4.
A Stage structured predator-prey dynamics
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where
τn (τ ) =
θ (τ ) + 2πn
,
ω(τ )
n = 0, 1, 2, . . . .
(4.6)
Implementation of the technique of Beretta and Kuang [4] to our system requires
that we have explicit expressions for the equilibrium components x ∗ and y ∗ . In the
remainder of this subsection we therefore concentrate on the case when p(x) = px,
p > 0 constant. In this case, the interior equilibrium is given by
dj τ
de
bpe−dj τ − d
∗ ∗
.
(x , y ) =
,
bp
bp 2 e−dj τ
The graphs shown in Fig. 2 are plots of the functions S0 (τ ) and S1 (τ ) defined above,
illustrating a case with dj > 0. The S0 curve shows that, for the parameter values
used (see caption) the interior equilibrium loses stability and then regains stability at
a larger τ , before the equilibrium itself finally disappears at τ = (1/dj ) ln(bp/d) ≈
3. The simulation results given by Fig. 3 confirm the stability findings provided by
Fig. 2.
Fig. 4 shows a plot of the function S0 (τ ) in a case when there is no juvenile
mortality (dj = 0), and should be compared with Fig. 2. There are two main differences, the first is that in Fig. 4 the interior equilibrium is feasible for all τ ≥ 0,
whereas in Fig. 2 it is feasible only up to a finite value of τ . The second difference
is that in Fig. 2 the equilibrium loses stability and then regains it as τ is increased,
whereas in Fig. 4 stability is permanently lost and, indeed, additional pairs of unstable eigenvalues appear as τ is further increased beyond the first bifurcation value.
This is an indication that one should expect more complicated dynamics, which is
what Fig. 5 does indeed show. The simulations shown in Fig. 5 should be compared
with the corresponding situation with juvenile mortality shown in Fig. 3. We shall
discuss the differences further in the final section of this paper.
5. Model implications
Lemma 1 implies that when the resource (x) is assumed to be constant, then the
consumer (y) population dynamics takes the simplest form of a globally attractive steady state. The linear stability analysis (Fig. 2) shows that if the resource is
dynamic, then there is a window in time delay parameter values that provides sustainable oscillatory dynamics. These observations suggest that it is very important
to model the resource dynamics explicitly. There are many popular mathematical
models in the literature that fail to do this and which lead to many misleading
statements. Examples of such popular models include the logistic equation and
the Lotka-Volterra competition model, where resources are assumed to be constant (critical comments can be found in Ginzburg [6], Turchin [21] and Kuang
et al. [15]).
In the special case when p(x) = px and τ = 0, it is easy to show that model (1.3)
has a globally attractive steady state in the positive quadrant. As the maturation time
delay τ increases, we see that oscillatory dynamics may appear and further increase
of τ will return the oscillatory dynamics to the globally attractive steady state form.
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S.A. Gourley, Y. Kuang
Graph of stability switch by S_0(T), S_1(T) and S_2(T), here T=tau
20
y
10
0
5
10
15
20
25
30
T
–10
–20
Fig. 4. Plot of the function S0 (τ ) when p = 1, b = 10, dj = 0 (i.e. no juvenile mortality)
and d = 0.5. The equilibrium is feasible for all τ ≥ 0. The first line (top one) depicts S0 (τ ),
the second line (middle one) depicts S1 (τ ), and the third line (bottom one) depicts S2 (τ ).
Each intersection point of these lines with the T = τ axis provides the threshold values
at which the number of roots with positive real parts of the characteristic equation about
the positive steady state jumps by 2. For example, when τ = 20, there are three pairs of
characteristic roots of the characteristic equation about the positive steady state that have
positive real parts.
Ultimately, when the maturation time delay is too long, the positive steady state
disappears and the consumer population dies out. This shows the sensitivity of the
model dynamics on maturation (through-stage) time delay. The ultimate scenario
makes intuitive biological sense: if it takes too long to mature then the juvenile
population will suffer low through-stage survival and, as a result, the highest possible recruitment rate to adulthood (be−dj τ p(1)) will drop below the adult death
rate d, leading to the extinction of y. However, in the unlikely case that juveniles
do not suffer any mortality (dj = 0), the oscillatory dynamics will persist and gain
complexity (in the sense that the number of characteristic roots with positive real
parts increases) when we increase the delay τ (see Figs. 4 and 5). Such distinct
dynamical outcomes highlight the importance of incorporating the through-stage
death rate in stage structured population models.
In the more realistic scenario when we assume that the functional response
function takes the Holling type II form p(x) = px/(1 + ax) and τ = 0, then
model (1.3) reduces to the well-known and well studied conventional Holling type
II predator-prey model (Kuang [13]). In this case, the nonlinearity of the functional
response alone can generate oscillatory dynamics.
We have made the assumption that only adult predators are capable of preying
on the prey species, and that the juveniles either live on their parents or subsist on
a resource that is different from that required by adults and the juvenile resource is
available in excess of requirements. This assumption is biologically restrictive. It
A Stage structured predator-prey dynamics
199
τ=0.1
τ=5
4
8
prey
predator
prey
predator
6
x, y
x, y
3
2
1
0
4
2
0
100
200
300
0
400
0
100
τ=15
200
8
prey
predator
prey
predator
6
x, y
6
x, y
400
τ=20
8
4
2
0
300
4
2
0
100
200
time t
300
400
0
0
100
200
time t
300
400
Fig. 5. A solution of model (1.3) with p(x) = px, x(θ) = 0.3, y(θ) = 1, θ ∈ [−τ, 0]
where p = 1.0, b = 10, dj = 0 (i.e. no juvenile mortality), d = 0.5 and τ varies from 0.25
to 25.
may hold for some carnivore-herbivore interactions such as bird-insect interactions,
but may fail to be true for most plant-herbivore interactions.
Acknowledgements. We would like to thank Professor Odo Diekmann and the two referees
for their many valuable suggestions that improved the presentation of this paper.
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