Journal of Mathematical Biology Manuscript-Nr. (will be inserted by hand later)

On periodic cohort solutions of a size-structured population model

Waltraud Huyer

Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria; E-mail address: huyer@cma.univie.ac.at

January 13, 1997

Abstract. We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n ≥ 2) with small size differences will tend to a periodic one-cohort solution as t → ∞.

Key words: Size-structured populations – Discontinuous reproduction – Variable environment – Periodic cohort solutions – Stability

1. Introduction

Reproduction is an event in the life of individuals. When there is quite some variation in the individual states at which such events occur, a formal appeal to the law of large numbers allows one to model reproduction by rates at the population level. When reproduction occurs upon reaching a specified individual state, one can still work with rates at the population level provided the distribution of the population with respect to individual state is smooth (i.e. absolutely continuous). However, when ‘giving birth’ is restricted to specific individual states and, in addition, the population is concentrated in some (moving) individual states, reproduction is an event even at the population level. The aim of this paper is to initiate the study of models incorporating these two features.

We consider a size-structured population model of this type, i.e. the state of an individual is characterised by its size. The individuals are born at size x_{α}, and when they reach size x_{ω} (x_{ω} > x_{α} > 0), they produce k offspring of size x_{α} and die (or, equivalently, split into k equal parts). There is feedback through the environment, i.e. the growth rate g depends on the food density S. The mortality rate μ is assumed to be a positive constant. Some information on suicide reproduction can be found in [5], and size-structured population models with binary fission into two equal or unequal parts are analysed in [6], [5], [4] and [3], for example.

We are interested in the special case that the population consists of finitely many cohorts. A cohort refers to a group of individuals of exactly the same state (i.e. size in the present context) and is mathematically described by a measure concentrated in a point, i.e. a Dirac measure. The underlying biology of single cohort solutions is very simple and therefore such models come natural to the biological mind. In models with sharply pulsed reproduction such as models for fish populations the population tends to be composed of distinguishable cohorts and periodic one-cohort solutions might be the dominating feature of the dynamics; cf. [7]. The full problem is described by an abstract renewal equation with a measure on [x_{α}, x_{ω}) as initial condition and a feedback law (cf. the ‘cumulative’ formulation of [1] and [2]), but in the case of finitely many cohorts it reduces to a system of coupled ordinary differential equations. If we start with n cohorts, we shall have n cohorts all the time. Let N_{i} denote the population number of cohort i and x_{i} the corresponding size. In the case of n cohorts the model equations read

\{ \begin{matrix}\frac{dN_{i}}{dt} = −μN_{i}, i = 1, …, n, \\ \frac{dx_{i}}{dt} = g(S, x_{i}), i = 1, …, n, \\ \frac{dS}{dt} = h(S, β) − ∑_{i=1}^{n} γ(S, x_{i})N_{i} \\ + initial conditions and jumps in N_{i} and x_{i} at times t \\ such that x_{i}(t) = x_{ω} for some i. \end{matrix}  (1.1)

The function h describes the dynamics of the unstructured variable S in the absence of a consumer, and γ is the consumption rate. We want to study the asymptotic behaviour and, in particular, the existence and stability of periodic n-cohort solutions. This behaviour might change if certain parameters in the model equations are varied. We assume that the function h also depends on a scalar parameter β that will be the bifurcation parameter for our problem. The following assumptions on the functions g, h and γ are used throughout this paper:

(H_{g}) g ∈ C([0, ∞) × [x_{α}, x_{ω}]; ℝ), for each x ∈ [x_{α}, x_{ω}] there exists an S_{x} ≥ 0 such that g(S, x) > 0 for S > S_{x} and g(S, x) = 0 for 0 ≤ S ≤ S_{x}, g is decreasing in x, strictly increasing in S on [S_{x}, ∞) for all x ∈ [x_{α}, x_{ω}] and twice continuously differentiable at those (S, x) for which g(S, x) > 0.

(H_{h}) h ∈ C^{1}([0, ∞) × (0, ∞); ℝ), h is strictly increasing in β, for each β > 0 there exists a unique S̃_{β} > 0 such that h(S̃_{β}, β) = 0 and \frac{∂h}{∂S} (S̃_{β}, β) =:

−d_{β} < 0.

(H_{γ}) γ ∈ C^{1}([0, ∞) × [x_{α}, x_{ω}]; ℝ), γ(0, x) = 0 for x ∈ [x_{α}, x_{ω}] and γ > 0 elsewhere, γ is increasing in S and x.

Moreover, we consider only values of β for which (H_{β}) S̃_{β} > S_{x_{ω}}.

The typical growth rates we have in mind are the so-called growth rates of von Bertalanffy type (cf. [6, p. 22] and [5], for example)

g(S, x) = c(f(S) − x)^{+},  (1.2)

where c > 0, z^{+} := max(0, z), f is a continuously differentiable, strictly increasing function of S with f(0) = 0 and lim_{S→∞} f(S) > x_{ω}, and x is of dimension ‘length’. In this case we have S_{x} = f^{−1}(x), and there is no growth if food becomes too scarce. The condition (H_{h}) implies that a larger value of β favours the abundance of food. We have h(S, β) > 0 for 0 < S < S̃_{β}, h(S, β) < 0 for S > S̃_{β} and S̃_{β} is strictly increasing in β. Moreover, S̃_{β} is the only stable biologically meaningful (i.e. nonnegative) equilibrium of S for fixed β in the absence of a consumer. Typical examples are h(S, β) = β − S (constant inflow of fresh, nonreproducing food particles combined with a constant food deterioration) and h(S, β) = S(1 − S/β) (logistic growth). The assumption (H_{γ}) reflects some biologically intuitive facts: if food is more abundant, more can be consumed (and in particular there cannot be any consumption in the absence of food), and larger individuals consume more. The condition (H_{β}) ensures that size x_{ω} is reached under the food conditions S ≡ S̃_{β}; otherwise size x_{ω} would never be reached for initial data S(0) ≤ S̃_{β} and the population would become extinct.

We consider n cohorts (that we assume to be arranged according to size; without loss of generality we can also assume that the first cohort is a cohort of newborns) and associate a mapping (discrete time dynamical system) with the ODE system. Let n ∈ ℕ, β > 0 with (H_{β}), x_{α} = x_{1,0} ≤ x_{2,0} ≤ … ≤ x_{n,0} ≤ x_{ω}, 0 < S_{0} ≤ S̃_{β}, N_{i,0} ≥ 0 for i = 1, …, n, and solve the equations

\{ \begin{matrix}\frac{dx_{i}}{dt} = g(S, x_{i}), x_{i}(0) = x_{i,0}, i = 1, …, n, \\ \frac{dS}{dt} = h(S, β) − e^{−μt} ∑_{i=1}^{n} γ(S, x_{i})N_{i,0}, S(0) = S_{0}. \end{matrix}  (1.3)

Remark 1.1. a) (H_{h}) implies that it is necessary to choose S_{0} ∈ (0, S̃_{β}] in order to possibly obtain a periodic solution.

b) We have to extend the mapping artificially to the case that the initial sizes can become x_{ω} in order to be able to deal with the degenerate case that x_{i,0} = x_{i+1,0} for some i. In this case, as well as in the case where N_{i,0} = 0 for some i, we actually have less than n cohorts.

We define a mapping F_{n} : ℝ^{2n} → ℝ^{2n}, (S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}) ↦ (S′_{0}, N′_{1,0}, …, N′_{n,0}, x′_{2,0}, …, x′_{n,0}) as follows. Let T = T (S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}) ≥ 0 be such that x_{n}(T) = x_{ω}, x_{n}(t) < x_{ω} for 0 ≤ t < T (if x_{n,0} = x_{ω}, we have T = 0), and set

\{ \begin{matrix}S′_{0} := S(T), \\ N′_{1,0} := kN_{n,0}e^{−μT} , \\ N′_{i,0} := N_{i−1,0}e^{−μT} , i = 2, …, n, \\ x′_{i,0} := x_{i−1}(T), i = 2, …, n. \end{matrix}  (1.4)

T is well-defined because of assumptions (H_{β}), (H_{h}) and (H_{γ}), and we have S′_{0} > 0, S′_{0} ∈ [S_{x_{ω}}, S̃_{β}], N_{i,0} ≥ 0 for i = 1, …, n and x_{α} ≤ x′_{2,0} ≤ … ≤ x′_{n,0} ≤ x_{ω}. The iterates of F_{n} give the food density, population numbers and corresponding sizes of the cohorts at the instants of reproduction, and a periodic point of F_{n} thus corresponds to a periodic n-cohort solution.

Remark 1.2. Since the size growth is not necessarily strictly monotonous, the mappings F_{n} might have discontinuities if S′_{0} = S_{x_{ω}} > 0, Ṡ(T −) < 0. (If S(T) = S_{x_{ω}}, Ṡ(T −) = 0, then S̈(T −) = μe^{−μT} ∑_{i=1}^{n} γ(S_{x_{ω}}, x_{i})N_{i,0} > 0, i.e. S would have a local minimum at t = T if there were no jumps, and this case is all right.)

In Sect. 2 the mapping F_{1} is investigated. We assume that the trivial fixed point (S̃_{β}, 0) becomes unstable at β = 1, and existence of a stable nontrivial fixed point of F_{1} (resp. a periodic one-cohort solution of Eq. (1.1)) is proved for 1 < β < β_{1} for some β_{1} > 1. In Sect. 3 a criterion is derived to determine whether for 1 < β < β_{1}, β sufficiently close to 1, the iterations of F_{2} will converge to the nontrivial fixed point (S̄_{0}(β), N̄_{0}(β)) of F_{1} for (S_{0}, N_{1,0} + N_{2,0}, x_{2,0}) sufficiently close to (S̄_{0}(β), N̄_{0}(β), x_{α}), i.e. whether the periodic one-cohort solution is stable with respect to splitting into two cohorts. The same criterion applies to splitting into n cohorts for all n ≥ 2, and the proof of this statement can be found in the Appendix. Moreover, local existence of nontrivial fixed points of F_{n} (i.e. periodic n-cohort solutions) is proved for n ≥ 2. In Sect. 4 we present some numerical examples that show that both stability and instability of the fixed point of F_{1} with respect to cohort splitting can occur in the case of linear growth rates (1.2).

2. The one-cohort problem

We consider now the special case n = 1, F_{1} =: F, and we drop the indices. Let β > 0 be such that (H_{β}) is satisfied. We have to solve the coupled ODE system

\{ \begin{matrix}\frac{dx}{dt} = g(S, x), x(0) = x_{α}, \\ \frac{dS}{dt} = h(S, β) − N_{0}e^{−μt}γ(S, x), S(0) = S_{0}. \end{matrix}  (2.1)

If T = T (S_{0}, N_{0}) > 0 is such that x(T) = x_{ω}, x(t) < x_{ω} for 0 ≤ t < T, the mapping F : (S_{0}, N_{0}) ↦ (S′_{0}, N′_{0}) is given by

\{ \begin{matrix}S′_{0} := S(T), \\ N′_{0} := kN_{0}e^{−μT} . \end{matrix}  (2.2)

Remark 2.1. We have T (S_{0}, N_{0}) > T (S_{0}, 0) for all S_{0} ∈ (0, S̃_{β}], N_{0} > 0 (if there is no consumption, the food density is larger and consequently the individuals reach x_{ω} more quickly), and S_{0} ↦ T (S_{0}, 0) is strictly decreasing on (0, S̃_{β}]. If

T (S̃_{β}, 0) ≥ \frac{1}{μ} ln k,

we have lim_{t→∞} N(t) = 0 and lim_{t→∞} S(t) = S̃_{β} for all choices of (S_{0}, N_{0}), and F has only the trivial fixed point (S_{0}, N_{0}) = (S̃_{β}, 0).

Notation 2.2. We introduce the abbreviations

T̃_{β} := T (S̃_{β}, 0)

and

T̄ := \frac{1}{μ} ln k,

i.e. T̄ is the time it takes to reduce the population number to one kth of its initial value if there is no reproduction. Note that T̃_{β} is strictly decreasing in β.

Remark 2.3. a) If the food density is constant S(t) ≡ S > S_{x_{ω}}, a population of newborns will reach size x_{ω} at time

t̄(S) := ∫_{x_{α}}^{x_{ω}} \frac{dx}{g(S, x)}.

The population will go extinct if t̄(S) < T̄ and increase unboundedly if t̄(S) > T̄. For t̄(S) = T̄ the number of offspring from the surviving members of an initial population N_{0} is just N_{0} and thus the population number is periodic with period T̄. The hypothesis (H_{g}) implies that t̄ is strictly decreasing on (S_{x_{ω}}, ∞) and therefore there exists at most one value S such that t̄(S) = T̄.

b) Remark 1.2 implies that discontinuities might occur if S(T) = S′_{0} = S_{x_{ω}} > 0, Ṡ(T −) < 0, i.e.

h(S_{x_{ω}}, β) − N_{0}e^{−μT} γ(S_{x_{ω}}, x_{ω}) < 0

or

N_{0} > \frac{e^{μT}}{γ(S_{x_{ω}}, x_{ω})}h(S_{x_{ω}}, β).

Since T > T̃_{β} for N_{0} > 0, the mapping is in particular continuously differentiable (differentiability of ODE solutions with respect to initial conditions and parameters) in the neighbourhood of (S_{0}, N_{0}) with S_{0} ∈ (0, S̃_{β}] and

0 ≤ N_{0} ≤ \frac{e^{μT̃_{β}}}{γ(S_{x_{ω}}, x_{ω})}h(S_{x_{ω}}, β).

At the points where F is differentiable we have

\{ \begin{matrix}\frac{∂S′_{0}}{∂S_{0}} = \frac{∂S}{∂S_{0}} (T) + Ṡ(T)\frac{∂T}{∂S_{0}} , \\ \frac{∂S′_{0}}{∂N_{0}} = \frac{∂S}{∂N_{0}} (T) + Ṡ(T)\frac{∂T}{∂N_{0}} , \\ \frac{∂N′_{0}}{∂S_{0}} = −μ\frac{∂T}{∂S_{0}} ke^{−μT} N_{0}, \\ \frac{∂N′_{0}}{∂N_{0}} = −μ\frac{∂T}{∂N_{0}} ke^{−μT} N_{0} + ke^{−μT} . \end{matrix}

We have to compute the eigenvalues of the Jacobi matrix of F in order to determine the stability of the trivial fixed point (0, S̃_{β}). To this end we first evaluate the derivatives of x and S with respect to initial data and parameters at (S_{0}, N_{0}) = (S̃_{β}, 0).

Lemma 2.4. Let (x, S) be the solution of Eq. (2.1), and let x̄ be the solution of

\frac{dx̄}{dt} = g(x̄, S̃_{β}), x̄(0) = x_{α}.

Then we have at (S_{0}, N_{0}) = (S̃_{β}, 0)

\frac{∂S}{∂S_{0}} (t) = e^{−d_{β}t},  (2.3)
\frac{∂x}{∂S_{0}} (t) = ∫_{0}^{t} exp (∫_{τ}^{t} \frac{∂g}{∂x}(S̃_{β}, x̄(σ)) dσ) \frac{∂g}{∂S} (S̃_{β}, x̄(τ))e^{−d_{β}τ} dτ, (2.4)

\frac{∂S}{∂N_{0}} (t) = −e^{−d_{β}t} ∫_{0}^{t} e^{(d_{β}−μ)τ} γ(S̃_{β}, x̄(τ)) dτ,  (2.5)

\frac{∂x}{∂N_{0}} (t) = ∫_{0}^{t} exp (∫_{τ}^{t} \frac{∂g}{∂x}(S̃_{β}, x̄(σ)) dσ) \frac{∂g}{∂S} (S̃_{β}, x̄(τ))\frac{∂S}{∂N_{0}} (τ) dτ, (2.6)
\frac{∂S}{∂β} (t) = \frac{1}{d_{β}} \frac{∂h}{∂β} (S̃_{β}, β)(1 − e^{−d_{β}t}),  (2.7)

\frac{∂x}{∂β} (t) = ∫_{0}^{t} exp (∫_{τ}^{t} \frac{∂g}{∂x}(S̃_{β}, x̄(σ)) dσ) \frac{∂g}{∂S} (S̃_{β}, x̄(τ))\frac{∂S}{∂β} (τ) dτ. (2.8)

Moreover we have at (S_{0}, N_{0}) = (S̃_{β}, 0)

\frac{∂T}{∂α} = −g(S̃_{β}, x_{ω})^{−1} \frac{∂x}{∂α}(T̃_{β})  (2.9)

for α = S_{0}, N_{0} and β.

Proof. Application of theorems about differentiability of ODE solutions with respect to initial data and parameters yields Eqs. (2.3)–(2.8), and by differentiating the relationship x(T) = x_{ω} we obtain Eq. (2.9). ⊓ ⊔

In particular the quantities (2.3)–(2.9) have the biologically obvious signs: the larger the initial value S_{0}, the larger S at later times (for N_{0} = 0) and consequently the individuals grow more quickly; a larger value of the bifurcation parameter β has a similar effect, but setting the initial population to a small positive value (and thus adding consumption to the food dynamics) has the opposite effect.

The following lemma shows that the stability of the trivial fixed point of F is as we might expect from Remarks 2.1 and 2.3, a), i.e., the population will go extinct if the unexploited equilibrium food density S̃_{β} is too small but that will not happen if S̃_{β} is large enough.

Lemma 2.5. The Jacobi matrix of F at the trivial fixed point (S_{0}, N_{0}) = (S̃_{β}, 0) reads

(DF)(S̃_{β}, 0) = ( \begin{matrix}e^{−d_{β} T̃_{β}} & \frac{∂S}{∂N_{0}} (T̃_{β}) \\ 0 & ke^{−μT̃_{β}} \end{matrix} )

and has the two real positive eigenvalues λ_{1} = e^{−d_{β} T̃_{β}} < 1 and λ_{2} = ke^{−μT̃_{β}}. The trivial fixed point is stable if λ_{2} < 1 or T̃_{β} > T̄ and unstable if λ_{2} > 1 or T̃_{β} < T̄, where T̃_{β} and T̄ are defined in Notation 2.2.

We want to discuss the existence of nontrivial fixed points of F for T̃_{β} < T̄. The mapping F can be extended to S_{0} > S̃_{β} and N_{0} < 0, but the case N_{0} < 0 has no biological meaning.

Definition 2.6. Let β_{0} ≥ 0 such that (H_{β}) is satisfied for β > β_{0}. Then we can define a mapping G : (0, ∞) × (−∞, ∞) × (β_{0}, ∞) → ℝ^{2} by

\{ \begin{matrix}G_{1}(S_{0}, N_{0}, β) := S(T) − S_{0}, \\ G_{2}(S_{0}, N_{0}, β) := ke^{−μT} − 1, \end{matrix}

where S is the solution of the second equation of (2.1) and T = T (S_{0}, N_{0}, β) is defined as before.

By Eq. (2.2) we must have G(S_{0}, N_{0}, β) = 0 if (S_{0}, N_{0}) is a nontrivial fixed point of F pertaining to the parameter value β, and in particular T (S_{0}, N_{0}, β) = T̄. Without loss of generality we assume that the trivial fixed point becomes unstable at β = 1, i.e.

T̃_{1} = T̄.  (2.10)

Since T̃_{1} = T (0, S̃_{1}) = ∫_{x_{α}}^{x_{ω}} \frac{dx}{g(S̃_{1},x)}, Remark 2.3, a) implies that S̃_{1} is just the

unique value such that t̄(S̃_{1}) = T̄. Moreover, we assume henceforth

d_{1} = 1,  (2.11)

which can be achieved by scaling the time variable and serves to simplify notation.

Theorem 2.7. Assume that (2.10) and (2.11) are satisfied. Then there exist a neighbourhood U of 1 in (0, ∞) and a neighbourhood V of (S̃_{1}, 0) in ℝ^{2} such that for each β ∈ U there exists a unique (S_{0}, N_{0}) ∈ V with G(S_{0}, N_{0}, β) = 0. Moreover, for β ∈ U with β < 1 we have S_{0} > S̃_{β}, N_{0} < 0, and for β ∈ U with β > 1 we have S_{0} < S̃_{β}, N_{0} > 0, i.e. only the second case yields a biologically meaningful nontrivial fixed point of F resp. a periodic one-cohort solution of Eq. (1.1).

Proof. By Eq. (2.10) we have T (S̃_{1}, 0, 1) = T̄ and G(S̃_{1}, 0, 1) = 0, and there exists a neighbourhood W of (S̃_{1}, 0, 1) such that G is continuously differentiable on W. Let y := (S_{0}, N_{0}). Then we have

(D_{y}G)(y, β) = ( \begin{matrix}\frac{∂S}{∂S_{0}} (T) + Ṡ(T)\frac{∂T}{∂S_{0}} − 1 & \frac{∂S}{∂N_{0}} (T) + Ṡ(T)\frac{∂T}{∂N_{0}} \\ −μke^{−μT} \frac{∂T}{∂S_{0}} & −μke^{−μT} \frac{∂T}{∂N_{0}} \end{matrix} ) ,

and in particular

(D_{y}G)(S̃_{1}, 0, 1) = ( \begin{matrix}k^{− \frac{1}{μ}} − 1 & \frac{∂S}{∂N_{0}} (T̄) \\ −μ\frac{∂T}{∂S_{0}} & −μ\frac{∂T}{∂N_{0}} \end{matrix} ) .  (2.12)

The determinant of this matrix is

Δ := det(D_{y}G)(S̃_{1}, 0, 1) = μ\frac{∂T}{∂N_{0}} (1 − k^{− \frac{1}{μ}}) + μ\frac{∂T}{∂S_{0}} \frac{∂S}{∂N_{0}} (T̄),  (2.13)

which is positive by Lemma 2.4 and assumptions (H_{g}) and (H_{γ}). The Implicit Function Theorem implies that there exist neighbourhoods U of 1 and V of (S̃_{1}, 0) and a C^{1}-function H : U → V such that V ×U ⊂ W and G(H(β), β) = 0 for all β ∈ U. Moreover, for β ∈ U the equation G(y, β) = 0 possesses one and only one solution y = H(β) in V.

Only three kinds of choices for (S_{0}, N_{0}) are possible such that S might return to its initial value: a) S_{0} > S̃_{β}, N_{0} < 0; b) S_{0} = S̃_{β}, N_{0} = 0; c) S_{0} < S̃_{β}, N_{0} > 0. Let β > 1. Then T (S_{0}, N_{0}) ≤ T̃_{β} < T̄ for S_{0} ≥ S̃_{β}, N_{0} ≤ 0, i.e. we must have case c). Analogously, case a) applies to β < 1. ⊓ ⊔

The fact that a nontrivial biologically meaningful fixed point of F can only exist for β > 1 can also be derived from the following heuristic considerations. If we add a very small number of individuals to a food population which is at its own dynamical equilibrium S̃_{β}, the dynamics of the consumer population will, initially at least, conform approximately to the assumption of a constant food density. Remark 2.3, a) and assumption (2.10) then tell us that the population will go extinct whenever S̃_{β} < S̃_{1} (resp. β < 1) and will start to grow when S̃_{β} > S̃_{1} (resp. β > 1).

The principle of exchange of stability (cf. [8, p. 93, Theorem 5.1], for example) implies

Theorem 2.8. Let the assumptions of Theorem 2.7 be satisfied. If U is chosen sufficiently small, the biologically meaningful nontrivial fixed point of F for β ∈ U ∩ (1, ∞) is stable.

We assume in the following that U is chosen sufficiently small such that the conclusion of Theorem 2.8 holds.

3. The n-cohort problem for n ≥ 2

In this section we address the following question: If we start with n ≥ 2 cohorts with small size differences, will these differences be larger after all n cohorts have reproduced, or will they be smaller and eventually tend to zero? Will the solution of the n-cohort problem converge to a periodic solution of the onecohort problem? This phenomenon cannot occur in the case of a constant food density or if the growth rate is independent of S since then the size differences always stay the same. First we consider the case n = 2, i.e. for (S_{0}, N_{1}, N_{2}, x_{0}) ∈ (0, S̃_{β}] × [0, ∞) × [0, ∞) × [x_{α}, x_{ω}], β > 0, we solve

\{ \begin{matrix}\frac{dx_{1}}{dt} = g(S, x_{1}), x_{1}(0) = x_{α}, \\ \frac{dx_{2}}{dt} = g(S, x_{2}), x_{2}(0) = x_{0}, \\ \frac{dS}{dt} = h(S, β) − e^{−μt}(N_{1}γ(S, x_{1}) + N_{2}γ(S, x_{2})), S(0) = S_{0}, \end{matrix}  (3.1)

where we again assume that (H_{β}), (2.10) and (2.11) are satisfied. The mapping F_{2} : (S_{0}, N_{1}, N_{2}, x_{0}) ↦ (S′_{0}, N′_{1}, N′_{2}, x′_{0}) is defined by

\{ \begin{matrix}S′_{0} := S(T), \\ N′_{1} := kN_{2}e^{−μT} , \\ N′_{2} := N_{1}e^{−μT} , \\ x′_{0} := x_{1}(T), \end{matrix}  (3.2)

where T = T (S_{0}, N_{1}, N_{2}, x_{0}) ≥ 0 is such that x_{2}(T) = x_{ω}, x_{2}(t) < x_{ω} for 0 ≤ t < T.

Notation 3.1. For β ∈ U we use the abbreviations

\{ \begin{matrix}S̄_{0}(β) := H_{1}(β), \\ N̄_{0}(β) := H_{2}(β), \end{matrix}

where H is defined by Theorem 2.7 (see the proof), i.e. (S̄_{0}(β), N̄_{0}(β)) is the nontrivial biologically meaningful fixed point of F_{1} for β ∈ U ∩(1, ∞). Moreover, we denote the solution of Eq. (2.1) corresponding to S_{0} = S̄_{0}(β), N_{0} = N̄_{0}(β) with (x̄_{β}, S̄_{β}).

The nontrivial fixed point (S̄_{0}(β), N̄_{0}(β)) of F_{1} for β ∈ U ∩ (1, ∞) can be viewed as a periodic point (S_{0}, N_{1}, N_{2}, x_{0}) of period two of F_{2} with S_{0} = S̄_{0}(β), N_{1} + N_{2} = N̄_{0}(β), x_{0} = x_{α} (or N_{1} + kN_{2} = N̄_{0}(β), x_{0} = x_{ω}), i.e. the artificial splitting of one cohort into two cohorts is not unique and instead of a fixed point of F_{1} we obtain a family of periodic points of period two of F_{2}. In order to investigate the ‘stability’ of these periodic points of period two, we have to consider the second iterate F_{2}^{2} =: K : (S_{0}, N_{1}, N_{2}, x_{0}) ↦ (S′′_{0}, N′′_{1}, N′′_{2}, x′′_{0}). If T_{1} := T (S_{0}, N_{1}, N_{2}, x_{0}) and T_{2} := T (F_{2}(S_{0}, N_{1}, N_{2}, x_{0})), we have N′′_{1} = kN_{1}e^{−μ(T_{1}+T_{2})} and N′′_{2} = kN_{2}e^{−μ(T_{1}+T_{2})}, i.e. the quotient \frac{N_{1}}{N_{2}} or \frac{N_{2}}{N_{1}} remains invariant under K if the denominator is not zero. (Biologically obvious: Since the mortality rate is constant and the individuals produce a fixed number of offspring, the ratios between the population numbers of the cohorts stay the same in later generations.) This allows us to restrict K to any set N_{2} = ηN_{1}, η a positive constant, and thus to reduce the problem by one dimension. Let K_{η} : (S_{0}, N_{0}, x_{0}) ↦ (S′′_{0}, N′′_{0}, x′′_{0}) be defined by

(S′′_{0}, \frac{1}{1 + η} N′′_{0}, \frac{η}{1 + η} N′′_{0}, x′′_{0}) := K(S_{0}, \frac{1}{1 + η} N_{0}, \frac{η}{1 + η} N_{0}, x_{0}).

The nontrivial fixed point (S̄_{0}(β), N̄_{0}(β)) of F_{1} for β ∈ U ∩(1, ∞) corresponds to the fixed point (S̄_{0}(β), N̄_{0}(β), x_{α}) (or (S̄_{0}(β), \frac{1+η}{1+kη} N̄_{0}(β), x_{ω})) of K_{η}. The Jacobi matrix of K_{η} at the point (S̄_{0}(β), N̄_{0}(β), x_{α}) is

(DK_{η})(S̄_{0}(β), N̄_{0}(β), x_{α}) = ( A \begin{matrix}∗ \\ ∗ \end{matrix}_{0 0 \frac{∂x′′_{0}}{∂x_{0}}} ) ,  (3.3)

where

A := (DF_{1})(S̄_{0}(β), N̄_{0}(β))  (3.4)

is the (2×2)-Jacobi matrix of F_{1} evaluated at its stable fixed point (S̄_{0}(β), N̄_{0}(β)) and ‘∗’ denotes a not so interesting number. Note that x′′_{0}(S_{0}, N_{0}, x_{α}) = x_{α} for all S_{0}, N_{0}, which accounts for the two zeros in the third row of the matrix (3.3).

Thus the eigenvalues of the Jacobi matrix (3.3) are given by the eigenvalues of A and

λ := \frac{∂x′′_{0}}{∂x_{0}} (S̄_{0}(β), \frac{1}{1 + η} N̄_{0}(β), \frac{η}{1 + η} N̄_{0}(β), x_{α}),  (3.5)

where x′′_{0} now again denotes the fourth component of the mapping K. The former are of absolute value < 1 by assumption, and the stability of the fixed point (S̄_{0}(β), N̄_{0}(β), x_{α}) of K_{η} depends on the value of λ, which has to be nonnegative since x′′_{0} > x_{α} for x_{0} > x_{α}.

Lemma 3.2. The eigenvalue λ defined by (3.5) is independent of η and given by

λ(β) = \frac{g(S̄_{0}(β), x_{α})}{g(S̄_{0}(β), x_{ω})} exp (∫ T̄ ; 0 \frac{∂g}{∂x}(S̄_{β}(t), x̄_{β}(t)) dt) ,  (3.6)

where S̄_{0}(β), S̄_{β}(t) and x̄_{β}(t) are defined by Notation 3.1 and T̄ is given by Notation 2.2. In particular, for a growth rate of the form (1.2) λ reads

λ(β) = \frac{f(S̄_{0}(β)) − x_{α}}{f(S̄_{0}(β)) − x_{ω}} e^{−cT̄} = \frac{f(S̄_{0}(β)) − x_{α}}{f(S̄_{0}(β)) − x_{ω}} k^{− \frac{c}{μ}} .  (3.7)

Proof. In order to compute λ, we first have to solve the following ODE system:

\{ \begin{matrix}\frac{dx_{1}}{dt} = g(S, x_{1}), x_{1}(0) = x_{α}, \\ \frac{dx_{2}}{dt} = g(S, x_{2}), x_{2}(0) = x_{0}, \\ \frac{dS}{dt} = h(S, β) − \frac{N̄_{0}(β)}{1 + η}e^{−μt}(γ(S, x_{1}) + ηγ(S, x_{2})), S(0) = S̄_{0}(β). \end{matrix} (3.8)

Let T_{1} ≥ 0 be such that x_{2}(T_{1}) = x_{ω}, x_{2}(t) < x_{ω} for 0 ≤ t < T_{1}. We set S′_{0} := S(T_{1}), x′_{0} := x_{1}(T_{1}), N′_{0} := e^{−μT_{1}} N̄_{0}(β) and solve

\{ \begin{matrix}\frac{dx̃_{1}}{dt} = g(S̃, x̃_{1}), x̃_{1}(0) = x_{α}, \\ \frac{dx̃_{2}}{dt} = g(S̃, x̃_{2}), x̃_{2}(0) = x′_{0}, \\ \frac{dS̃}{dt} = h(S̃, β) − \frac{N′_{0}}{1 + η} e^{−μt}(kηγ(S̃, x̃_{1}) + γ(S̃, x̃_{2})), S̃(0) = S′_{0}. \end{matrix}

Let T_{2} ≥ 0 be such that x̃_{2}(T_{2}) = x_{ω}, x̃_{2}(t) < x_{ω} for 0 ≤ t < T_{2}, and define x′′_{0} := x̃_{1}(T_{2}). Then x′′_{0} is differentiable with respect to x_{0} for 0 ≤ x_{0} − x_{α} < ε, ε > 0 suitably chosen, where the derivative is replaced with the right-hand derivative at x_{0} = x_{α}. For x_{0} = x_{α} we have T_{1} = T̄, T_{2} = 0, S′_{0} = S̄_{0}(β), x′_{0} = x_{ω}, and we obtain

\frac{dx′′_{0}}{dx_{0}} (x_{α}) = ˙ x̃_{1}(0)\frac{dT_{2}}{dx_{0}} (x_{α}) = g(S̄_{0}(β), x_{α})\frac{dT_{2}}{dx_{0}} (x_{α}),

where we have used \frac{∂x̃_{1}}{∂x′_{0}} (0) = \frac{∂x̃_{1}}{∂S′_{0}} (0) = \frac{∂x̃_{1}}{∂N′_{0}} (0) = 0. Differentiation of the relationship x̃_{2}(T_{2}) = x_{ω} with respect to x_{0} yields at x_{0} = x_{α}

˙x̃_{2}(0)\frac{dT_{2}}{dx_{0}} (x_{α}) + \frac{∂x̃_{2}}{∂x′_{0}} (0)\frac{dx′_{0}}{dx_{0}} (x_{α}) = g(S̄_{0}(β), x_{ω})\frac{dT_{2}}{dx_{0}} (x_{α}) + \frac{dx′_{0}}{dx_{0}} (x_{α}) = 0,

and we obtain

\frac{dT_{2}}{dx_{0}} (x_{α}) = −\frac{1}{g(S̄_{0}(β), x_{ω})} \frac{dx′_{0}}{dx_{0}} (x_{α})

and

λ = \frac{dx′′_{0}}{dx_{0}} (x_{α}) = −\frac{g(S̄_{0}(β), x_{α})}{g(S̄_{0}(β), x_{ω})} \frac{dx′_{0}}{dx_{0}} (x_{α}).

We have

\frac{dx′_{0}}{dx_{0}} = \frac{∂x_{1}}{∂x_{0}} (T_{1}) + ẋ_{1}(T_{1})\frac{dT_{1}}{dx_{0}} ,

and for x_{0} = x_{α} we obtain similarly as above, using the definition of T_{1} and the fact that T_{1}(x_{α}) = T̄,

\frac{dx′_{0}}{dx_{0}} (x_{α}) = \frac{∂x_{1}}{∂x_{0}} (T̄) − \frac{∂x_{2}}{∂x_{0}} (T̄).

If we differentiate the solutions of Eq. (3.8) with respect to the initial condition x_{0} at x_{0} = x_{α}, we obtain the differential equations

\{ \begin{matrix}\frac{d}{dt} \frac{∂x_{1}}{∂x_{0}} = \frac{∂g}{∂x}(S̄_{β}, x̄_{β})\frac{∂x_{1}}{∂x_{0}} + \frac{∂g}{∂S} (S̄_{β}, x̄_{β})\frac{∂S}{∂x_{0}} , \frac{∂x_{1}}{∂x_{0}} (0) = 0, \\ \frac{d}{dt} \frac{∂x_{2}}{∂x_{0}} = \frac{∂g}{∂x}(S̄_{β}, x̄_{β})\frac{∂x_{2}}{∂x_{0}} + \frac{∂g}{∂S} (S̄_{β}, x̄_{β})\frac{∂S}{∂x_{0}} , \frac{∂x_{2}}{∂x_{0}} (0) = 1, \end{matrix}

which implies that z(t) := \frac{∂x_{2}}{∂x_{0}} (t) − \frac{∂x_{1}}{∂x_{0}} (t) is the solution of the equation

\frac{dz}{dt} = \frac{∂g}{∂x}(S̄_{β}, x̄_{β})z, z(0) = 1.

From this we infer

\frac{∂x_{2}}{∂x_{0}} (t) − \frac{∂x_{1}}{∂x_{0}} (t) = exp (∫_{0}^{t} \frac{∂g}{∂x}(S̄_{β}(τ), x̄_{β}(τ)) dτ) ,

and Eqs. (3.6) and (3.7) follow. ⊓ ⊔

Remark 3.3. The eigenvalue λ(β) given by Eq. (3.6) can be biologically interpreted as the ratio between the flux of individuals from size x_{α} at t = 0 and the influx to size x_{ω} at time t = T̄ if the food density is given by S(t) = S̄_{β}(t) for 0 ≤ t ≤ T̄ and mortality is ignored. This makes biological sense: if the influx to x_{ω} at t = T̄ is larger than the flux from x_{α} at t = 0, the individuals of the first cohort can only grow from x_{α} to x′′_{0} < x_{0} while the individuals of the second cohort grow from x′_{0} to x_{ω} (in the notation of the proof of Lemma 3.2) for 0 < x_{0} − x_{α} infinitesimally small. This corresponds to the case of an eigenvalue λ(β) ∈ (0, 1) and stability of the periodic one-cohort solution with respect to splitting into two cohorts. Likewise the case λ(β) > 1 has a biologically meaningful interpretation.

The equations \frac{dx̄_{β}}{dt} = g(S̄_{β}, x̄_{β}), x̄_{β}(0) = x_{α} and x̄_{β}(T̄) = x_{ω} yield

\frac{g(S̄_{0}(β), x_{α})}{g(S̄_{0}(β), x_{ω})} = exp (− ∫ T̄ ; 0 \frac{d}{dt} ln g(S̄_{0}(β), x̄_{β}(t)) dt)

= exp (− ∫ T̄ ; 0 \frac{∂g}{∂x}(S̄_{0}(β), x̄_{β}(t))\frac{g(S̄_{β}(t), x̄_{β}(t))}{g(S̄_{0}(β), x̄_{β}(t))} dt) ,

and we obtain the useful representation

λ(β) = exp (∫ T̄ ; 0 (\frac{∂g}{∂x}(S̄_{β}(t), x̄_{β}(t)) − \frac{∂g}{∂x}(S̄_{0}(β), x̄_{β}(t))\frac{g(S̄_{β}(t), x̄_{β}(t))}{g(S̄_{0}(β), x̄_{β}(t))}) dt) .

(3.9)

For β = 1 we have by assumption (2.10) S̄_{1}(t) ≡ S̄_{0}(1) = S̃_{1}, and Eq. (3.9) immediately implies

λ(1) = 1.

In particular we obtain for a linear growth rate from Eq. (3.7)

\frac{f(S̃_{1}) − x_{α}}{f(S̃_{1}) − x_{ω}} k^{− \frac{c}{μ}} = 1.  (3.10)

Corollary 3.4. a) If in addition to (H_{g}) we have

S ↦ \frac{∂}{∂x}(ln g(S, x)) is strictly increasing in S for S > S_{x}, x ∈ [x_{α}, x_{ω}], (3.11)

and

S̄_{0}(β) = max_{t≥0} S̄_{β}(t)  (3.12)

for a β ∈ U ∩ (1, ∞), then the nontrivial fixed point (S̄_{0}(β), N̄_{0}(β), x_{α}) of K_{η} is stable for all η > 0.

b) In case of a linear growth rate (1.2) the nontrivial fixed point (S̄_{0}(β), N̄_{0}(β), x_{α}) of K_{η} is stable for all η > 0 for a β ∈ U ∩ (1, ∞) if and only if S̄_{0}(β) > S̃_{1}.

Proof. The assertion a) follows from the representation (3.9) and the assumptions (H_{g}), (3.11) and (3.12).

In the case of a linear growth rate (1.2), Eqs. (3.7) and (3.10) and the strict monotony of f imply that λ(β) < 1 if and only if S̄_{0}(β) > S̃_{1}, which proves statement b). ⊓ ⊔

Corollary 3.4, b) says that for von Bertalanffy growth rates (1.2) the nontrivial periodic one-cohort solution is stable with respect to splitting into two cohorts if and only if the value of the food density is ‘large enough’ (namely larger than S̃_{1}) at the instants of reproduction.

Remark 3.5. In numerical computations of nontrivial periodic one-cohort solutions for growth rates (1.2), linear h, consumption rates as in Sect. 4 and conservation of mass (cf. Sect. 4) S did attain its maximum at the instants of reproduction. However, we do not know if the property (3.12) was a just a peculiarity of our examples or if it may also occur for growth rates that satisfy (3.11) but are not of type (1.2).

We have sign(λ(β) − 1) = sign λ′(1) for β > 1 sufficiently close to 1, and the stability of the fixed point (S̄_{0}(β), N̄_{0}(β), x_{α}) of K_{η} depends on the sign of λ′(1). In order to compute λ′(1), we need the following lemma.

Lemma 3.6. Let (x, S) be the solution of Eq. (2.1), let T be defined as in Sect. 2, and let S̄_{0}(β) and N̄_{0}(β) be defined by Notation 3.1. Then the derivatives with respect to β at β = 1 are given by

\frac{d}{dβ} S̄_{0}(1) = \frac{μ}{Δ} ((1 − k^{− \frac{1}{μ}})\frac{∂h}{∂β} (S̃_{1}, 1)\frac{∂T}{∂N_{0}} − \frac{∂T}{∂β} \frac{∂S}{∂N_{0}} (T̄))  (3.13)

and

\frac{d}{dβ} N̄_{0}(1) = −\frac{μ}{Δ}(1 − k^{− \frac{1}{μ}}) (\frac{∂h}{∂β} (S̃_{1}, 1)\frac{∂T}{∂S_{0}} + \frac{∂T}{∂β} ) ,  (3.14)

where Δ is defined by Eq. (2.13), T̄ is defined by Notation 2.2 and all partial derivatives are evaluated at (S_{0}, N_{0}, β) = (S̃_{1}, 0, 1).

Proof. If G is given by Definition 2.6 and H and U are defined as in the proof of Theorem 2.7, differentiation of the relationship G(H(β), β) = 0, β ∈ U, with respect to β yields

(D_{y}G)(H(β), β)\frac{dH}{dβ} (β) + \frac{∂G}{∂β}(H(β), β) = 0, β ∈ U.

Since the matrix (D_{y}G)(S̃_{1}, 0, 1) (cf. (2.12)) is invertible, we obtain in particular

\frac{dH}{dβ} (1) = ( \begin{matrix}\frac{d}{dβ} S̄_{0}(1) \\ \frac{d}{dβ} N̄_{0}(1) \end{matrix} ) = −((D_{y}G)(S̃_{1}, 0, 1))^{−1} \frac{∂G}{∂β}(H(1), 1)

= −\frac{1}{Δ} ( \begin{matrix}−μ\frac{∂T}{∂N_{0}} & −\frac{∂S}{∂N_{0}} (T̄) \\ μ\frac{∂T}{∂S_{0}} & k^{− \frac{1}{μ}} − 1 \end{matrix} ) ( \begin{matrix}\frac{∂S}{∂β} (T̄) \\ −μ\frac{∂T}{∂β} \end{matrix} ) ,

where all derivatives are evaluated at (S_{0}, N_{0}, β) = (S̃_{1}, 0, 1). From Eq. (2.7) and assumption (2.11) we obtain \frac{∂S}{∂β} (T̄) = \frac{∂h}{∂β} (S̃_{1}, 1)(1 − k^{− \frac{1}{μ}}), which finishes the proof. ⊓ ⊔

Eq. (3.14), the representations of the derivatives in Lemma 2.4 and the assumptions immediately imply \frac{d}{dβ} N̄_{0}(1) > 0, which reflects the fact that we have N̄_{0}(β) > 0 for β ∈ U ∩ (1, ∞) and N̄_{0}(β) < 0 for β ∈ U ∩ (0, 1). The sign of \frac{d}{dβ} S̄_{0}(β), however, is a more complicated matter; cf. Lemma 3.9.

Lemma 3.7. λ′(1) is given by

λ′(1) = −\frac{μ}{Δ} (\frac{∂T}{∂β} + \frac{∂h}{∂β} (S̃_{1}, 1)\frac{∂T}{∂S_{0}} ) (\frac{∂S}{∂N_{0}} (T̄) ∫ T̄ ; 0 \frac{\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t))}{g(S̃_{1}, x̄_{1}(t))}e^{−t} dt

+ (1 − k^{− \frac{1}{μ}}) ∫ T̄ ; 0 \frac{\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t))}{g(S̃_{1}, x̄_{1}(t))} (\frac{∂S}{∂N_{0}} (t) + e^{−μt}γ(S̃_{1}, x̄_{1}(t))) dt) (3.15)

and

sign λ′(1) = sign (\frac{∂S}{∂N_{0}} (T̄) ∫ T̄ ; 0 \frac{\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t))}{g(S̃_{1}, x̄_{1}(t))}e^{−t} dt

+ (1 − k^{− \frac{1}{μ}}) ∫ T̄ ; 0 \frac{\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t))}{g(S̃_{1}, x̄_{1}(t))} (\frac{∂S}{∂N_{0}} (t) + e^{−μt}γ(S̃_{1}, x̄_{1}(t))) dt) ,

(3.16)

where all partial derivatives are evaluated at (S_{0}, N_{0}, β) = (S̃_{1}, 0, 1).

Proof. We differentiate Eq. (3.9) at β = 1, and by integration by parts we obtain

\begin{aligned}λ′(1) & = & ∫ T̄ ; 0 (\frac{∂^{2}g}{∂S∂x}(S̃_{1}, x̄_{1}(t)) − \frac{∂g}{∂x}(S̃_{1}, x̄_{1}(t)) \frac{\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t))}{g(S̃_{1}, x̄_{1}(t))}) \\ & & × (\frac{dS̄_{β}}{dβ} |_{β=1}(t) − \frac{d}{dβ} S̄_{0}(1)) dt \\ & = & ∫ T̄ ; 0 \frac{d}{dt} ( \frac{\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t))}{g(S̃_{1}, x̄_{1}(t))}) (\frac{dS̄_{β}}{dβ} |_{β=1}(t) − \frac{d}{dβ} S̄_{0}(1)) dt \\ & = & − ∫ T̄ ; 0 \frac{\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t))}{g(S̃_{1}, x̄_{1}(t))} \frac{d}{dt} (\frac{dS̄_{β}}{dβ} |_{β=1}(t)) dt, \end{aligned}  (3.17)

where we have used λ(1) = 1 and ˙ x̄_{1}(t) = g(S̃_{1}, x̄_{1}(t)). We have

\frac{dS̄_{β}}{dβ} |_{β=1}(t) = \frac{∂S}{∂β} (t) + \frac{∂S}{∂S_{0}} (t)\frac{d}{dβ} S̄_{0}(1) + \frac{∂S}{∂N_{0}} (t)\frac{d}{dβ} N̄_{0}(1),

where all partial derivatives are evaluated at (S_{0}, N_{0}, β) = (S̃_{1}, 0, 1), and Eqs. (2.3), (2.5), (2.7) and (2.11) yield

\frac{d}{dt} (\frac{dS̄_{β}}{dβ} (t)) = (\frac{∂h}{∂β} (S̃_{1}, 1) − \frac{d}{dβ} S̄_{0}(1)) e^{−t}
− \frac{d}{dβ} N̄_{0}(1) (\frac{∂S}{∂N_{0}} (t) + e^{−μt}γ(S̃_{1}, x̄_{1}(t))) . (3.18)

Equation (3.13) and the definition of Δ (cf. Eq. (2.13)) imply

\frac{∂h}{∂β} (S̃_{1}, 1) − \frac{d}{dβ} S̄_{0}(1) = \frac{μ}{Δ} (\frac{∂h}{∂β} (S̃_{1}, 1)\frac{∂T}{∂S_{0}} + \frac{∂T}{∂β} ) \frac{∂S}{∂N_{0}} (T̄),  (3.19)

and by using Eqs. (3.18), (3.19) and (3.14) in Eq. (3.17) we obtain Eq. (3.15). Equations (2.4) and (2.7)–(2.9) and assumptions (H_{g}) and (H_{h}) yield

\begin{matrix}\frac{∂T}{∂β} + \frac{∂h}{∂β} (S̃_{1}, 1)\frac{∂T}{∂S_{0}} \\ = −g(S̃_{1}, x_{ω})^{−1} \frac{∂h}{∂β} (S̃_{1}, 1) ∫ T̄ ; 0exp(∫ T̄ ; t \frac{∂g}{∂x}(S̃_{1}, x̄_{1}(τ)) dτ)\frac{∂g}{∂S} (S̃_{1}, x̄_{1}(t)) dt < 0, \end{matrix}

from which assertion (3.16) follows. ⊓ ⊔ Remark 3.8. The sign of λ′(1) does not depend on the particular shape of h but only on the value of its positive zero at β = 1.

For a growth rate of the form (1.2) we obtain the following result; cf. Corollary 3.4, b).

Lemma 3.9. In the special case of a linear growth rate (1.2) we have sign λ′(1) = −sign \frac{d}{dβ} S̄_{0}(1) and

\frac{d}{dβ} S̄_{0}(1) = \frac{μ}{Δ}(f(S̃_{1}) − x_{ω})^{−1}f′(S̃_{1})\frac{∂h}{∂β} (S̃_{1}, 1)k^{− \frac{c}{μ}}

×((1 − k^{− \frac{1}{μ}}) ∫ T̄ ; 0 e^{(c−1)t} ∫_{0}^{t} e^{(1−μ)τ} γ(S̃_{1}, x̄_{1}(τ)) dτ dt

−k^{− \frac{1}{μ}} ∫ T̄ ; 0 e^{ct}(1 − e^{−t}) dt ∫ T̄ ; 0 e^{(1−μ)t}γ(S̃_{1}, x̄_{1}(t)) dt),

and consequently

sign λ′(1) = sign(k^{− \frac{1}{μ}} ∫ T̄ ; 0 e^{ct}(1 − e^{−t}) dt ∫ T̄ ; 0 e^{(1−μ)t}γ(S̃_{1}, x̄_{1}(t)) dt

− (1 − k^{− \frac{1}{μ}}) ∫ T̄ ; 0 e^{(c−1)t} ∫_{0}^{t} e^{(1−μ)τ} γ(S̃_{1}, x̄_{1}(τ)) dτ dt).

Proof. Eq. (3.7) and the monotony of f imply sign λ′(1) = −sign\frac{d}{dβ} S̄_{0}(1), and we have \frac{∂g}{∂x}(S, x) = −c and \frac{∂g}{∂S} (S, x) = cf′(S) at those (S, x) for which f(S) > x. From Eqs. (2.5)–(2.9) we obtain at (S_{0}, N_{0}, β) = (S̃_{1}, 0, 1)

\frac{∂T}{∂N_{0}} = \frac{f′(S̃_{1})}{f(S̃_{1}) − x_{ω}} e^{−cT̄} ∫ T̄ ; 0 e^{(c−1)t} ∫_{0}^{t} e^{(1−μ)τ} γ(S̃_{1}, x̄_{1}(τ)) dτ dt

and

\frac{∂T}{∂β} = −\frac{∂h}{∂β} (S̃_{1}, 1)\frac{f′(S̃_{1})}{f(S̃_{1}) − x_{ω}} e^{−cT̄} ∫ T̄ ; 0 e^{ct}(1 − e^{−t}) dt,

and Eqs. (3.13), (2.5) and the assumptions on f and h yield the assertions. ⊓ ⊔

Biological intuition immediately tells us that a nontrivial periodic one-cohort solution is stable with respect to splitting into n cohorts (n ≥ 2) if and only if this is the case for n = 2. Mathematical details are deferred to the Appendix, and we obtain the following result, which shows that the nontrivial periodic one-cohort solution is stable with respect to splitting into n cohorts for β > 1 sufficiently close to 1 if λ′(1) < 0.

Theorem 3.10. Let λ′(1) < 0. Then there exists a β_{1} > 1 such that for each β ∈ (1, β_{1}) there exist a neighbourhood V_{β} of (S̄_{0}(β), N̄_{0}(β)) and an

ε_{β} > 0 such that lim_{k→∞} F_{n}^{nk}(S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}) = (S̄_{0}(β), \frac{N_{1,0}N̄_{0}(β)}{∑_{i=1}^{n} N_{i,0}} ,

…, \frac{N_{n,0}N̄_{0}(β)}{∑_{i=1}^{n} N_{i,0}} , x_{α}, …, x_{α}) for (S_{0}, ∑_{i=1}^{n} N_{i,0}) ∈ V_{β}, 0 ≤ x_{i,0} − x_{α} < ε_{β} for

i = 2, …, n and all n ≥ 2.

No general statement can be made about sign λ′(1); cf. Sect. 4.

We conclude this section with an investigation of fixed points of F_{n} for n ≥ 2. For a fixed point (S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}) of F_{n} we must have N_{1,0} = ke^{−μT} N_{n,0} and N_{i,0} = e^{−μT} N_{i−1,0}, i = 2, …, n, which implies that either N_{i,0} = 0 for i = 1, …, n or

T = \frac{1}{nμ} ln k = \frac{T̄}{n}  (3.20)

and

N_{i,0} = k^{−(i−1)/n}N_{1,0}, i = 2, …, n,  (3.21)

where T̄ is defined by Notation 2.2. For values of β such that (H_{β}) is satisfied we define the function A_{β} : [x_{α}, x_{ω}] → ℝ by

A_{β}(x) := ∫_{x_{α}}^{x} \frac{dσ}{g(S̃_{β}, σ)}, x ∈ [x_{α}, x_{ω}].

A_{β}(x) is the time it takes an individual to grow from size x_{α} to x or, equivalently, the age of an individual of size x if the food density is S̃_{β}. Naturally, A_{β} is strictly increasing on [x_{α}, x_{ω}], A_{β}(x_{α}) = 0, A_{β}(x_{ω}) = T̃_{β}, which implies that A_{β}^{−1} : [0, T̃_{β}] → [x_{α}, x_{ω}] exists. The point (S̃_{β}, 0, …, 0, x_{2,0}, …, x_{n,0}) (no population present and the food at its unexploited equilibrium) is a periodic point of period ≤ n of F_{n} for all choices of x_{i,0}, i = 2, …, n. It is a fixed point of F_{n} if and only if

x_{i,0} = A_{β}^{−1}(\frac{i−1}{n} T̃_{β}), i = 2, …, n,

i.e., if the initial sizes are chosen such that the time it takes to grow from x_{i,0} to x_{i+1,0} (x_{1,0} := x_{α}) is the same for i = 1, …, n − 1 and equals the time it takes to grow from x_{n,0} to x_{ω} under food conditions S ≡ S̃_{β}, and this time is necessarily T = \frac{1}{n} T̃_{β}.

In order to investigate the existence of nontrivial fixed points of F_{n}, n ≥ 2, we define a mapping similarly to Definition 2.6 and again call it G.

Definition 3.11. Let W ⊂ ℝ^{n+1} an appropriately chosen neighbourhood of (S̃_{1}, 0, x̄_{2,0}, …, x̄_{n,0}, 1), where

x̄_{i,0} := A_{1}^{−1}(\frac{i−1}{n} T̄), i = 2, …, n.  (3.22)

For (S_{0}, N_{0}, x_{2,0}, …, x_{n,0}, β) ∈ W we solve the ODE system

\{ \begin{matrix}\frac{dx_{1}}{dt} = g(S, x_{1}), x_{1}(0) = x_{α}, \\ \frac{dx_{i}}{dt} = g(S, x_{i}), x_{i}(0) = x_{i,0}, i = 2, …, n, \\ \frac{dS}{dt} = h(S, β) − N_{0}e^{−μt} ∑_{i=1}^{n} k^{−(i−1)/n}γ(S, x_{i}), S(0) = S_{0}. \end{matrix}  (3.23)

Let T ≥ 0 be such that x_{n}(T) = x_{ω}, x_{n}(t) < x_{ω} for 0 ≤ t < T. Then we define G : W → ℝ^{n} by

\{ \begin{matrix}G_{1}(S_{0}, N_{0}, x_{2,0}, …, x_{n,0}, β) := S(T) − S_{0}, \\ G_{2}(S_{0}, N_{0}, x_{2,0}, …, x_{n,0}, β) := ke^{−nμT} − 1, \\ G_{i}(S_{0}, N_{0}, x_{2,0}, …, x_{n,0}, β) := x_{i−2}(T) − x_{i−1,0}, i = 3, …, n + 1. \end{matrix} (3.24)

If W is chosen sufficiently small, G is continuously differentiable in W, and Eqs. (2.10) and (3.22) imply G(S̃_{1}, 0, x̄_{2,0}, …, x̄_{n,0}, 1) = 0. In view of Eqs. (3.20) and (3.21) (S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}) is a nontrivial fixed point of F_{n} if and only if N_{i,0} = k^{−(i−1)/n}N_{0}, i = 1, …, n, and G(S_{0}, N_{0}, x_{2,0}, …, x_{n,0}, β) = 0.

We want to apply the Implicit Function Theorem to the mapping G. To this end we first compute the derivatives of the solutions of the ODE system (3.23) with respect to initial data and parameters.

Lemma 3.12. Let (x_{1}, …, x_{n}, S) be the solution of Eq. (3.23), let T be defined as in Definition 3.11 and T̄ by Notation 2.2, and let

x̄_{i}(t) := A_{1}^{−1}(t + \frac{i−1}{n} T̄), 0 ≤ t ≤ \frac{n−i+1}{n} T̄, i = 1, …, n,

i.e. x̄_{i}(t) is the size at time t of an individual with initial size x̄_{i,0} if the food density is S ≡ S̃_{1}. Then the partial derivatives at (S_{0}, N_{0}, x_{2,0}, …, x_{n,0}, β) = (S̃_{1}, 0, x̄_{2,0}, …, x̄_{n,0}, 1) are given by

\frac{∂S}{∂S_{0}} (t) = e^{−t},  (3.25)

\frac{∂x_{i}}{∂S_{0}} (t) = ∫_{0}^{t} exp (∫_{τ}^{t} \frac{∂g}{∂x}(S̃_{1}, x̄_{i}(σ)) dσ) \frac{∂g}{∂S} (S̃_{1}, x̄_{i}(τ))e^{−τ} dτ, i = 1, …, n,

(3.26)

\frac{∂S}{∂x_{j,0}} (t) = 0, j = 2, …, n,  (3.27)
\frac{∂x_{i}}{∂x_{j,0}} (t) = 0, i = 1, …, n, j = 2, …, n, j ≠ i,  (3.28)

\frac{∂x_{i}}{∂x_{i,0}} (t) = exp (∫_{0}^{t} \frac{∂g}{∂x}(S̃_{1}, x̄_{i}(τ)) dτ) , i = 2, …, n,  (3.29)

\frac{∂S}{∂N_{0}} (t) = −e^{−t} ∑_{i=1}^{n} k^{−(i−1)/n} ∫_{0}^{t} e^{(1−μ)τ} γ(S̃_{1}, x̄_{i}(τ)) dτ,  (3.30)

\frac{∂x_{i}}{∂N_{0}} (t) = ∫_{0}^{t} exp (∫_{τ}^{t} \frac{∂g}{∂x}(S̃_{1}, x̄_{i}(σ)) dσ) \frac{∂g}{∂S} (S̃_{1}, x̄_{i}(τ))\frac{∂S}{∂N_{0}} (τ) dτ,

i = 1, …, n,  (3.31)
\frac{∂T}{∂α} = −g(S̃_{1}, x_{ω})^{−1} \frac{∂x_{n}}{∂α} (\frac{1}{n}T̄) for α = S_{0}, N_{0} and x_{i,0}, i = 2, …, n, (3.32)

and in particular

\frac{∂T}{∂x_{j,0}} = 0, j = 2, …, n − 1.  (3.33)

The following theorem is an analogue to Theorem 2.7 for F_{n}, n ≥ 2. It shows that for each n ≥ 2 there exists, for β > 1 sufficiently close to 1, a nontrivial periodic n-cohort solution whose period is just one nth of that of the periodic one-cohort solution. Moreover, the individuals of cohort i reach the initial body size of the individuals of cohort i + 1, i = 1, …, n − 1, at the instance when the individuals of cohort n reproduce.

Theorem 3.13. Let n ≥ 2. There exists a β_{1} > 1 such that for each β ∈ (1, β_{1}) there exists a fixed point (S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}), N_{i,0} = k^{−(i−1)/n}N_{0}, i = 1, …, n, of F_{n} with N_{0} > 0.

Proof. We apply the Implicit Function Theorem to the function G given by Definition 3.11. Let y := (S_{0}, N_{0}, x_{2,0}, …, x_{n,0}) and ȳ := (S̃_{1}, 0, x̄_{2,0}, …, x̄_{n,0}). Then the Jacobi matrix (D_{y}G)(ȳ, 1) reads

( \begin{matrix}\frac{∂G_{1}}{∂S_{0}} & \frac{∂G_{1}}{∂N_{0}} & 0 & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & 0 \\ \frac{∂G_{2}}{∂S_{0}} & \frac{∂G_{2}}{∂N_{0}} & 0 & ⋯ & ⋯ & ⋯ & ⋯ & 0 & \frac{∂G_{2}}{∂x_{n,0}} \\ \frac{∂G_{3}}{∂S_{0}} & \frac{∂G_{3}}{∂N_{0}} & −1 & 0 & ⋯ & ⋯ & ⋯ & 0 & \frac{∂G_{3}}{∂x_{n,0}} \\ \frac{∂G_{4}}{∂S_{0}} & \frac{∂G_{4}}{∂N_{0}} & \frac{∂G_{4}}{∂x_{2,0}} & −1 & ... & & & ⋮ & ⋮ \\ \frac{∂G_{5}}{∂S_{0}} & \frac{∂G_{5}}{∂N_{0}} & 0 & \frac{∂G_{5}}{∂x_{3,0}} & ... & ... & & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & ... & ... & ... & ... & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & & ... & ... & ... & 0 & \frac{∂G_{n−1}}{∂x_{n,0}} \\ ⋮ & ⋮ & ⋮ & & & ... & ... & −1 & \frac{∂G_{n}}{∂x_{n,0}} \\ \frac{∂G_{n+1}}{∂S_{0}} & \frac{∂G_{n+1}}{∂N_{0}} & 0 & ⋯ & ⋯ & ⋯ & 0 & \frac{∂G_{n+1}}{∂x_{n−1,0}} & \frac{∂G_{n+1}}{∂x_{n,0}} \end{matrix} )

since Eqs. (3.24), (3.27), (3.28) and (3.33) imply \frac{∂G_{1}}{∂x_{i,0}} = 0, i = 2, …, n, \frac{∂G_{2}}{∂x_{i,0}} =

0, i = 2, …, n − 1, \frac{∂G_{i+1}}{∂x_{i,0}} = −1, i = 2, …, n − 1, and \frac{∂G_{i}}{∂x_{j,0}} = 0, i = 3, …, n, j ≠ i − 2, i − 1, n at (y, β) = (ȳ, 1). After a tedious but elementary computation we obtain

det(D_{y}G)(ȳ, 1) = (−1)^{n} ∑_{i=3}^{n+1} (\frac{∂G_{1}}{∂S_{0}} (\frac{∂G_{2}}{∂N_{0}} \frac{∂G_{i}}{∂x_{n,0}} − \frac{∂G_{2}}{∂x_{n,0}} \frac{∂G_{i}}{∂N_{0}} )

− \frac{∂G_{1}}{∂N_{0}} (\frac{∂G_{2}}{∂S_{0}} \frac{∂G_{i}}{∂x_{n,0}} − \frac{∂G_{2}}{∂x_{n,0}} \frac{∂G_{i}}{∂S_{0}} )) ∏_{j=i−1}^{n−1} \frac{∂G_{j+2}}{∂x_{j,0}} .

Differentiation of Eq. (3.24) at (y, β) = (ȳ, 1) yields

\frac{∂G_{2}}{∂N_{0}} \frac{∂G_{i}}{∂x_{n,0}} − \frac{∂G_{2}}{∂x_{n,0}} \frac{∂G_{i}}{∂N_{0}} = nμ (\frac{∂T}{∂N_{0}} δ_{i,n+1} + \frac{∂T}{∂x_{n,0}} \frac{∂x_{i−2}}{∂N_{0}} (T̄_{n}))

and

\frac{∂G_{2}}{∂S_{0}} \frac{∂G_{i}}{∂x_{n,0}} − \frac{∂G_{2}}{∂x_{n,0}} \frac{∂G_{i}}{∂S_{0}} = nμ (\frac{∂T}{∂S_{0}} δ_{i,n+1} + \frac{∂T}{∂x_{n,0}} \frac{∂x_{i−2}}{∂S_{0}} (T̄_{n}))

for i = 3, …, n+1, where we have used Eq. (2.10), T̄_{n} := \frac{1}{nμ} ln k and δ_{ij} denotes the Kronecker delta, and further

det(D_{y}G)(ȳ, 1) = (−1)^{n}nμ(− (1 − \frac{∂S}{∂S_{0}} (T̄_{n})) \frac{∂T}{∂N_{0}} − \frac{∂S}{∂N_{0}} (T̄_{n})\frac{∂T}{∂S_{0}}

+ ∑_{i=3}^{n+1} \frac{∂T}{∂x_{n,0}} (− (1 − \frac{∂S}{∂S_{0}} (T̄_{n})) \frac{∂x_{i−2}}{∂N_{0}} (T̄_{n})

− \frac{∂S}{∂N_{0}} (T̄_{n})\frac{∂x_{i−2}}{∂S_{0}} (T̄_{n})) ∏_{j=i−1}^{n−1} \frac{∂x_{j}}{∂x_{j,0}} (T̄_{n})) .

From Eqs. (3.25), (3.26) and (3.29)–(3.32) and assumptions (H_{g}) and (H_{γ}) we infer \frac{∂S}{∂S_{0}} (T̄_{n}) < 1, \frac{∂S}{∂N_{0}} (T̄_{n}) < 0, \frac{∂T}{∂S_{0}} < 0, \frac{∂T}{∂N_{0}} > 0, \frac{∂T}{∂x_{n,0}} < 0, \frac{∂x_{i−2}}{∂S_{0}} (T̄_{n}) > 0,

\frac{∂x_{i−2}}{∂N_{0}} (T̄_{n}) < 0, i = 3, …, n + 1, and \frac{∂x_{j}}{∂x_{j,0}} (T̄_{n}) > 0, j = 2, …, n, which implies det(D_{y}G)(ȳ, 1) ≠ 0 and sign(det(D_{y}G)(ȳ, 1)) = (−1)^{n−1}. Thus we can apply the Implicit Function Theorem to G and obtain a nontrivial biologically meaningful fixed point of F_{n} for β ∈ (1, β_{1}), β_{1} > 1 suitably chosen. ⊓ ⊔

4. Numerical results

In the first example we investigate sign λ′(1) numerically for the case of a linear growth rate (1.2) and a particular kind of consumption rate γ.

Example 4.1. We consider a linear growth rate (1.2) and scale the size variable such that x_{α} = 1. Let the consumption rate be given by (cf. [6, p. 21])

γ(S, x) = f_{1}(S)x^{2},  (4.1)

where f_{1} is a strictly increasing function of S with f_{1}(0) = 0, and assume that the conditions (2.10) and (2.11) are satisfied, i.e.

\frac{1}{μ} ln k = \frac{1}{c} ln \frac{f(S̃_{1}) − 1}{f(S̃_{1}) − x_{ω}}

or

f(S̃_{1}) = \frac{k^{\frac{c}{μ}} x_{ω} − 1}{k^{\frac{c}{μ}} − 1}.

Then we have

x̄_{1}(t) = \frac{k^{\frac{c}{μ}} x_{ω} − 1}{k^{\frac{c}{μ}} − 1} − e^{−ct} \frac{k^{\frac{c}{μ}} (x_{ω} − 1)}{k^{\frac{c}{μ}} − 1}.

By Lemma 3.9 the stability of the nontrivial fixed point of the one-cohort problem within the set of n-cohort solutions depends on the sign of the function

q(c, μ, k, x_{ω}) := k^{− \frac{1}{μ}} ∫_{0}^{\frac{1}{μ} ln k} e^{ct}(1 − e^{−t}) dt

× ∫_{0}^{\frac{1}{μ} ln k} e^{(1−μ)t}(\frac{k^{\frac{c}{μ}} x_{ω} − 1}{k^{\frac{c}{μ}} − 1} − e^{−ct} \frac{k^{\frac{c}{μ}} (x_{ω} − 1)}{k^{\frac{c}{μ}} − 1})^{2} dt

−(1 − k^{− \frac{1}{μ}}) ∫_{0}^{\frac{1}{μ} ln k} e^{(c−1)t} ∫_{0}^{t} e^{(1−μ)τ} (\frac{k^{\frac{c}{μ}} x_{ω} − 1}{k^{\frac{c}{μ}} − 1} − e^{−cτ} \frac{k^{\frac{c}{μ}} (x_{ω} − 1)}{k^{\frac{c}{μ}} − 1})^{2}dτdt,

where c > 0, μ > 0, k > 1 and x_{ω} > 1 and these parameters are in general not connected by any condition and cannot be normalised by any additional scaling.

First we considered the case kx_{α}^{3} = k = x_{ω}^{3} (conservation of mass). The function q was computed for a grid of (c, μ, x_{ω}) ∈ [0.001, 100] × [0.001, 100] × [1.25, 4.65] (i.e. the cases k = 2 up to k = 100 are included in this set) and the function q was always negative, which means that the fixed point of the one-cohort problem is stable for F_{n}, n ≥ 2.

Next we let k = 2 fixed (division into two parts) and computed q for a grid of (c, μ, x_{ω}) ∈ [0.001, 100] × [0.001, 100] × [ ^{3} \sqrt{2}, 10] (i.e. kx_{α}^{3} ≤ x_{ω}^{3}). There exists an increasing function μ_{1} : [0.001, 100] → [0.001, 10] such that for (c, μ) ∈ [0.001, 100] × [0.001, 100] with μ ≥ μ_{1}(c) there exists an x_{1}(c, μ) ∈ [ ^{3} \sqrt{2}, 10] such that q(c, μ, x_{ω}) ≥ 0 for x_{1}(c, μ) ≤ x_{ω} ≤ 10 and q negative elsewhere on [0.001, 100] × [0.001, 100] × [ ^{3} \sqrt{2}, 10]. The function x_{1} is decreasing in μ and increasing in c. For other values of k we obtained similar results, i.e. positive values of q could only occur for kx_{α}^{3} < x_{ω}^{3}.

The mapping F_{n}, n ≤ 10, was iterated numerically for linear growth rates (1.2) and consumption rates (4.1) with

f(S) = \frac{c_{1}S}{c_{2}S + 1}, f_{1}(S) = \frac{c_{3}S}{c_{2}S + 1},  (4.2)

where c_{1}, c_{2} and c_{3} are positive constants (Holling type II functional response, cf. [6]), and linear or logistic h for kx_{α}^{3} = x_{ω}^{3} (i.e. we considered cases where λ′(1) < 1 according to Example 4.1), β > 1 and different values of the initial data and the parameters. In all of our examples the iterations converged to a nontrivial fixed point of F_{1} even if the initial data were not chosen close to this fixed point.

In the following example we investigate the asymptotic behaviour numerically for a case where the fixed point of the one-cohort problem is not stable for F_{n}, n ≥ 2.

Example 4.2. We consider a linear growth rate (1.2) and a consumption rate (4.1) with (4.2), where c = 1.5, c_{1} = c_{3} = 1, c_{2} = 0.4, and let μ = 0.5, k = 2, x_{α} = 1, x_{ω} = 2 and h(S, β) = 15β − S. The trivial fixed point of F_{1} becomes unstable at β = 1 and we have q(1.5, 0.5, 2, 2) > 0 (i.e. we chose the parameters such that q is positive). The nontrivial fixed point of F_{1} was computed for β = \frac{16}{15}, \frac{17}{15}, \frac{6}{5}, \frac{19}{15} and \frac{4}{3} (cf. Table 1). The function S̄_{0} is strictly decreasing on [1, \frac{4}{3}].

Table 1. \begin{matrix}β & S̄_{0}(β) & N̄_{0}(β) \\ \frac{16}{15} & 14.97 & 0.2392 \\ \frac{17}{15} & 14.94 & 0.4782 \\ \frac{6}{5} & 14.92 & 0.7170 \\ \frac{19}{15} & 14.89 & 0.9556 \\ \frac{4}{3} & 14.86 & 1.1939 \end{matrix}

For β = \frac{19}{15} we investigated the asymptotic behaviour of F_{n} numerically for n = 2, 3, 4. Note that the quotients N_{i,0}/N_{j,0} remain invariant under F_{n}^{n}.

For n = 2 we have after one application of F_{2} N′_{1,0}/N′_{2,0} = kN_{2,0}/N_{1,0}. We iterated F_{2} for different start values and the iterations converged to different periodic points of period two, depending on the start value of N_{1,0}/N_{2,0}. In the case that we started with N_{1,0}/N_{2,0} = \sqrt{k} the periodic point of period two degenerated to the fixed point (14.95, 0.4847, 0.3427, 1.739) of F_{2}. In Table 2 we show some periodic points of period two of F_{2}. Similarly periodic points of period n were found by iterating F_{n} for n = 3, 4, and these periodic points can again degenerate to fixed points. By replacing h by h(S, β) = S(1 − \frac{S}{15β}) we obtained similar results.

Table 2. \begin{matrix}S_{0} & N_{1,0} & N_{2,0} & x_{2,0} \\ 15.09 & 0.0814 & 0.8145 & 1.222 \\ 14.89 & 0.8748 & 0.0437 & 1.966 \\ 15.07 & 0.2960 & 0.5921 & 1.330 \\ 14.88 & 0.6632 & 0.1658 & 1.942 \\ 15.06 & 0.3534 & 0.5301 & 1.383 \\ 14.88 & 0.6074 & 0.2025 & 1.928 \\ 15.03 & 0.4345 & 0.4345 & 1.515 \\ 14.89 & 0.5304 & 0.2652 & 1.883 \end{matrix}

Example 4.2 suggests that in the ‘unstable’ case (λ′(1) > 1) there exist families of periodic points of period n of F_{n}, n ≥ 2, for 1 < β < β_{1} that can be ‘degenerate’ (i.e. periodic points of period m, 1 ≤ m < n, m a divisor of n) and belong to the domain of attraction of F_{n}.

5. Concluding remarks

This simple population model exhibits the feature that cohorts ‘synchronise’ in the course of time at least for particular choices of the functions, parameters and initial values. This behaviour is well-documented by numerical results. It still remains to determine the domain of attraction in the ‘unstable’ case (λ′(1) > 0). We conjecture that in that case there are families of periodic points of period n (that can be ‘degenerate’, i.e. periodic points of period m, m a divisor of n) of F_{n} for all n belonging to the domain of attraction of F_{n}. Moreover, the results we obtained are only local in nature. Since the mappings F_{n} might have discontinuities and it is not easy to quantify where they occur, it is difficult to make global statements about the existence and stability of periodic points. However, if S_{x} = 0 for all x ∈ [x_{α}, x_{ω}] in the condition (H_{g}), there are no discontinuities.

The robustness of the results with respect to slight changes in the model specification is also still an open problem. For example, it is only an idealisation to say that all individuals reproduce at size x_{ω} when experimentally they are observed to reproduce at an almost fixed size. When reproduction occurs at a stochastically predetermined size, independent of any characteristics of an individual’s forbears, provided it does not die before that time, single cohort solutions do not exist any more and we have to analyse the model by means of the associated renewal equation. We conjecture that, if the ‘size at reproduction precluding death’ is only a little distributed around x_{ω}, there exists a periodic solution with one peak instead of a periodic single cohort solution, and this solution will be stable with respect to little perturbations in the initial conditions if λ(β) < 1 for the associated model with reproduction at size x_{ω}, i.e. if the periodic one-cohort solution of that model is stable with respect to splitting into n cohorts. A few simple numerical experiments seem to support this conjecture.

6. Appendix: Proof of Theorem 3.10 for n ≥ 3

Let n ≥ 3. The nontrivial fixed point (S̄_{0}(β), N̄_{0}(β)) of F_{1} for β ∈ U ∩ (1, ∞) corresponds to a family of periodic points (S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}), S_{0} = S̄_{0}(β), ∑_{i=1}^{n} N_{i,0} = N̄_{0}(β), x_{i,0} = x_{α} for i = 2, …, n, of period n of F_{n}. Let F_{n}^{n} : (S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0}) ↦ (S_{0}^{(n)}, N_{1,0}^{(n)}, …, N_{n,0}^{(n)}, x_{2,0}^{(n)}, …, x_{n,0}^{(n)}) and T_{i} := T (F_{n}^{i−1}(S_{0}, N_{1,0}, …, N_{n,0}, x_{2,0}, …, x_{n,0})), i = 1, …, n. Then N_{i,0}^{(n)} =

kN_{i,0}e^{−μ ∑_{j=1}^{n} T_{j}}, i = 1, …, n, so the quotients N_{i,0}/N_{j,0} remain invariant under F_{n}^{n} if the denominators are not zero. For a vector η = (η_{2}, …, η_{n}), η_{i} > 0 for i = 2, …, n, we define K_{η} : (S_{0}, N_{0}, x_{2,0}, …, x_{n,0}) ↦ (S_{0}^{(n)}, N_{0}^{(n)}, x_{2,0}^{(n)}, …, x_{n,0}^{(n)}) by

(S_{0}^{(n)}, \frac{1}{ζ}N_{0}^{(n)}, \frac{η_{2}}{ζ} N_{0}^{(n)}, …, \frac{η_{n}}{ζ} N_{0}^{(n)}, x_{2,0}^{(n)}, …, x_{n,0}^{(n)})

:= F_{n}^{n}(S_{0}, \frac{1}{ζ}N_{0}, \frac{η_{2}}{ζ} N_{0}, …, \frac{η_{n}}{ζ} N_{0}, x_{2,0}, …, x_{n,0}),

where ζ := 1 + ∑_{i=2}^{n} η_{i}. The Jacobi matrix of K_{η} at the nontrivial fixed point (S̄_{0}(β), N̄_{0}(β), x_{α}, …, x_{α}) equals

( \begin{matrix}A & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}⋯ \\ ⋯ \end{matrix} & \begin{matrix}⋯ \\ ⋯ \end{matrix} & \begin{matrix}⋯ \\ ⋯ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 & 0 & \frac{∂x_{2,0}^{(n)}}{∂x_{2,0}} & 0 & ⋯ & ⋯ & 0 \\ ⋮ & ⋮ & 0 & \frac{∂x_{3,0}^{(n)}}{∂x_{3,0}} & ... & ⋮ \\ ⋮ & ⋮ & ⋮ & ... & ... & ... & ⋮ \\ ⋮ & ⋮ & ⋮ & ... & ... & 0 \\ 0 & 0 & 0 & ⋯ & ⋯ & 0 & \frac{∂x_{n,0}^{(n)}}{∂x_{n,0}} \end{matrix} ) ,  (6.1)

where the (2 × 2)-matrix A is again defined by Eq. (3.4) and the zeros in the last (n − 1) rows are due to the fact that x_{i,0}^{(n)} = x_{α} for x_{i,0} = x_{α} independent of S_{0}, N_{0} and x_{j,0}, j ≠ i, for i = 2, …, n.

The following lemma is a generalisation of Lemma 3.2 to the case n ≥ 3.

Lemma 6.1. We have

\frac{∂x_{i,0}^{(n)}}{∂x_{i,0}} (S̄_{0}(β), N̄_{0}(β), x_{α}, …, x_{α}) = λ(β), i = 2, …, n,

where λ is given by Eq. (3.6).

Proof. Let n ≥ 3. The quantities x_{i,0}^{(n)} for S_{0} = S̄_{0}(β), N_{0} = N̄_{0}(β) are obtained by solving the following recursive ODE systems. First solve the system

\{ \begin{matrix}\frac{dx_{1}^{(1)}}{dt} = g(S^{(1)}, x_{1}^{(1)}), x_{1}^{(1)}(0) = x_{α}, \\ \frac{dx_{i}^{(1)}}{dt} = g(S^{(1)}, x_{i}^{(1)}), x_{i}^{(1)}(0) = x_{i,0}, i = 2, …, n, \\ \frac{dS^{(1)}}{dt} = h(S^{(1)}, β) − N̄_{0}(β)ζ^{−1}e^{−μt} ∑_{i=1}^{n} η_{i}γ(S^{(1)}, x_{i}^{(1)}), S^{(1)}(0) = S̄_{0}(β), \end{matrix}

where η_{1} := 1 and ζ := ∑_{i=1}^{n} η_{i} as before. Let T_{1} ≥ 0 be such that x_{n}^{(1)}(T_{1}) = x_{ω}, x_{n}^{(1)}(t) < x_{ω} for 0 ≤ t < T_{1}, and set S_{0}^{(1)} := S^{(1)}(T_{1}), N_{0}^{(1)} := N̄_{0}(β)e^{−μT_{1}} and x_{i,0}^{(1)} := x_{i−1}^{(1)}(T_{1}), i = 2, …, n.

For j = 2, …, n we solve recursively the ODE systems

\{ \begin{matrix}\frac{dx_{1}^{(j)}}{dt} = g(S^{(j)}, x_{1}^{(j)}), x_{1}^{(j)}(0) = x_{α}, \\ \frac{dx_{i}^{(j)}}{dt} = g(S^{(j)}, x_{i}^{(j)}), x_{i}^{(j)}(0) = x_{i,0}^{(j−1)}, i = 2, …, n, \\ \frac{dS^{(j)}}{dt} = h(S^{(j)}, β) − N_{0}^{(j−1)}ζ^{−1}e^{−μt}(k ∑_{i=1}^{j−1} η_{n+i−j+1}γ(S^{(j)}, x_{i}^{(j)}) \\ + ∑_{i=j}^{n} η_{i−j+1}γ(S^{(j)}, x_{i}^{(j)})), S^{(j)}(0) = S_{0}^{(j−1)}. \end{matrix}

Let T_{j} ≥ 0 be such that x_{n}^{(j)}(T_{j}) = x_{ω}, x_{n}^{(j)}(t) < x_{ω} for 0 ≤ t < T_{j}, and define S_{0}^{(j)} := S^{(j)}(T_{j}), N_{0}^{(j)} := N_{0}^{(j−1)}e^{−μT_{j}}, x_{i,0}^{(j)} := x_{i−1}^{(j)}(T_{j}), i = 2, …, n.

For (x_{2,0}, …, x_{n,0}) = (x_{α}, …, x_{α}) we have T_{1} = T̄ and T_{j} = 0 for j = 2, …, n. Moreover, for j = 1, …, n we have x_{i,0}^{(j)} = x_{α}, i = 2, …, j, and x_{i,0}^{(j)} = x_{ω}, i = j + 1, …, n. We shall in the following often use the fact that

\frac{∂x_{i}^{(j)}}{∂x_{i,0}^{(j−1)}} (0) = 1, i = 2, …, n,  (6.2)

and

\frac{∂x_{i}^{(j)}}{∂N_{0}^{(j−1)}} (0) = \frac{∂x_{i}^{(j)}}{∂S_{0}^{(j−1)}} (0) = \frac{∂x_{i}^{(j)}}{∂x_{k,0}^{(j−1)}} (0) = 0, k ≠ i, i = 1, …, n, (6.3)

for j = 2, …, n.

For the remainder of the proof we assume that all partial derivatives are taken at (x_{2,0}, …, x_{n,0}) = (x_{α}, …, x_{α}). By differentiating the equations x_{n}^{(j)}(T_{j}) = x_{ω}, j = 1, …, n, and by using Eqs. (6.2) and (6.3) we obtain for i = 2, …, n

\frac{∂T_{1}}{∂x_{i,0}} = −\frac{1}{g(S̄_{0}(β), x_{ω})} \frac{∂x_{n}^{(1)}}{∂x_{i,0}} (T̄)

and

\frac{∂T_{j}}{∂x_{i,0}} = −\frac{1}{g(S̄_{0}(β), x_{ω})} \frac{∂x_{n,0}^{(j−1)}}{∂x_{i,0}} , j = 2, …, n.

For j = 2, …, n, i = 2, …, n we have

\frac{∂x_{2,0}^{(j)}}{∂x_{i,0}} = ẋ_{1}^{(j)}(0)\frac{∂T_{j}}{∂x_{i,0}} = −\frac{g(S̄_{0}(β), x_{α})}{g(S̄_{0}(β), x_{ω})} \frac{∂x_{n,0}^{(j−1)}}{∂x_{i,0}}  (6.4)

and

\frac{∂x_{k,0}^{(j)}}{∂x_{i,0}} = ẋ_{k−1}^{(j)}(0)\frac{∂T_{j}}{∂x_{i,0}} + \frac{∂x_{k−1,0}^{(j−1)}}{∂x_{i,0}} = \frac{∂x_{k−1,0}^{(j−1)}}{∂x_{i,0}} − \frac{g(S̄_{0}(β), x_{k−1,0}^{(j−1)})}{g(S̄_{0}(β), x_{ω})} \frac{∂x_{n,0}^{(j−1)}}{∂x_{i,0}}

= \{ \begin{matrix}\frac{∂x_{k−1,0}^{(j−1)}}{∂x_{i,0}} − \frac{g(S̄_{0}(β), x_{α})}{g(S̄_{0}(β), x_{ω})} \frac{∂x_{n,0}^{(j−1)}}{∂x_{i,0}} , & k = 3, …, j, \\ \frac{∂x_{k−1,0}^{(j−1)}}{∂x_{i,0}} − \frac{∂x_{n,0}^{(j−1)}}{∂x_{i,0}} , & k = j + 1, …, n. \end{matrix}  (6.5)

The derivatives of x_{k,0}^{(1)} are given by

\frac{∂x_{k,0}^{(1)}}{∂x_{i,0}} = ẋ_{k−1}^{(1)}(T̄)\frac{∂T_{1}}{∂x_{i,0}} + \frac{∂x_{k−1}^{(1)}}{∂x_{i,0}} (T̄) = \frac{∂x_{k−1}^{(1)}}{∂x_{i,0}} (T̄) − \frac{∂x_{n}^{(1)}}{∂x_{i,0}} (T̄), k, i = 2, …, n.

(6.6)

From Eqs. (6.4)–(6.6) we obtain recursively

\frac{∂x_{i,0}^{(n)}}{∂x_{i,0}} = −\frac{g(S̄_{0}(β), x_{α})}{g(S̄_{0}(β), x_{ω})} (\frac{∂x_{1}^{(1)}}{∂x_{i,0}} (T̄) − \frac{∂x_{i}^{(1)}}{∂x_{i,0}} (T̄)) , i = 2, …, n.

Similarly as in the proof of Lemma 3.2 we have

\frac{∂x_{i}^{(1)}}{∂x_{i,0}} (T̄) − \frac{∂x_{1}^{(1)}}{∂x_{i,0}} (T̄) = exp (− ∫ T̄ ; 0 \frac{∂g}{∂x}(S̄_{β}(t), x̄_{β}(t)) dt) , i = 2, …, n,

which finishes the proof. ⊓ ⊔

Thus the eigenvalues of the Jacobi matrix (6.1) are the eigenvalues of the matrix A and the (n − 1)-fold eigenvalue λ, which implies that the stability of the nontrivial periodic one-cohort solution with respect to splitting into n cohorts (n ≥ 3) also only depends on the value of the single quantity λ(β).

Acknowledgement. Part of this work was done during a ten-month stay at the Centrum voor Wiskunde en Informatica (CWI) in Amsterdam supported by NUFFIC. I would like to thank the CWI for its hospitality. Moreover, I am grateful to Odo Diekmann for stimulating discussions and valuable suggestions, which considerably helped to improve an earlier version of this paper. Finally, thanks go to an anonymous referee for his detailed and useful comments.

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