TOPOLOGICAL CHAOS

FOR BIRTH–AND–DEATH–TYPE MODELS WITH PROLIFERATION

JACEK BANASIAK

School of Mathematical and Statistical Sciences, University of Natal, Durban 4041, South Africa

MIROSŁAW LACHOWICZ

Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, PL–02-097 Warsaw, Poland

Abstract—In this paper we consider a class of infinite systems of linear ODEs. Each system corresponds to a process characterized by two components: the conservative one (birth–and–death process) and the proliferative one. The system of this type can describe the population of neoplastic cells divided into subpopulations characterized by different levels of cellular resistance to antineoplastic drugs. Under suitable assumptions on the “birth” (amplification) coefficients, the “death” (deamplification) coefficients and the averages of the life-spans we prove that this class of models is topologically chaotic.

Key words— topological chaos, birth–and–death process, hypercyclic operators, kinetic model, cancer cells.

AMS Subject Classification— 34G10, 47D06, 47N60, 95C50, 92D25

1. Introduction

In the recent decade it has been observed that chaos, typically associated with nonlinear phenomena, can occur also in linear infinite–dimensional systems. To make this statement more precise, let us recall that the (semi–)dynamical system {T (t)}_{t≥0} on a metric space X is said to be topologically chaotic (chaotic in the sense of Devaney [7]) if

it is topologically transitive, i.e. for any two non-empty open sets U, V ⊂ X there exists t_{0} ≥ 0 such that T (t_{0})U ∩ V ≠ ∅

and its set of periodic points is dense in X.

Historically speaking, the original Devaney’s definition contained also the intuitive notion of sensitive dependence on initial conditions (sdic) but it was proved in [4] that in fact the two conditions given above yield sdic (the original proof for discrete in time systems can be repeated with slight modifications for continuous in time systems — see [1]). A link between chaos theory and linear operator theory has been established when it was realized that in a separable Banach space the topological transitivity was equivalent to the notion of hypercyclicity (see e.g. [6], [11]):

a strongly continuous semigroup of bounded linear operators {T (t)}_{t≥0} on a Banach space X is called hypercyclic if for some x ∈ X the orbit {T (t)x}_{t≥0} is dense in X.

This path was followed later in [12] (for discrete in time systems in Hilbert spaces) and several other papers (cf. [1]).

The significance of the notion of linear chaos for the applied mathematics (and continuous in time systems) was probably first noted in [17] where the authors showed that a class of linear systems (with constant coefficients) of kinetic type that can model e.g. the death process, scattering with energy loss, and many other phenomena, is chaotic. However, it should be mentioned that in order to be chaotic, the system must contain a “creation” (proliferative) part, as no conservative system (such as e.g. the standard death or birth–and–death process) can be chaotic in the whole Banach space.

The paper [17] was followed by [21] where it was shown that a linear transport equation modelling certain cell population dynamics is chaotic.

The recent paper [6] summarizes and considerably extends many of the existing results. The authors show, among others, that the convection-diffusion on the halfline is chaotic, rendering thus another important example of linear chaotic system. They also provide two important theorems allowing the identification of certain classes of chaotic and non-chaotic systems. As these two theorems will be the main tools in our investigations here, we shall quote them for readers’ convenience.

Theorem 1.1 ([6] — Theorem 3.1). Let X be a separable Banach space and A be the infinitesimal generator of a strongly continuous semigroup {T (t)}_{t≥0} on X. Suppose that the point spectrum σ_{P} (A), of the operator A, is such that there exists a nonempty open set U ⊂ σ_{P} (A) with U ∩ iℝ^{1} ≠ ∅. Assume that for each λ ∈ U there exists a corresponding nonzero eigenvector x_{λ} such that for any 0 ≠ Φ ∈ X^{∗} the function λ → F_{Φ}(λ) = ⟨Φ, x_{λ}⟩ is analytic on U and does not vanish identically on U. Then{T (t)}_{t≥0} is topologically chaotic.

Theorem 1.2 ([6] — Theorem 3.3). Let {T (t)}_{t≥0} be a hypercyclic semigroup generated by A in a Banach space X. Then the adjoint A^{∗} of A and the dual semigroup {T^{∗}(t)}_{t≥0} have the following properties:

(i) for any 0 ≠ ϕ ∈ X^{∗} the orbit {T^{∗}(t)ϕ}_{t≥0} is unbounded; (ii) the point spectrum σ_{P} (A^{∗}) is empty.

A survey of chaotic models coming from the applied sciences together with some discussion of links between discrete and continuous time theory can be found in [1].

The result of [17] was extended in [3] to cover a class of models with variable coefficients showing that the property of being chaotic is in a certain sense stable.

On the other hand, also in [3], it was proved that the “opposite” class of birth models cannot be chaotic.

In the present paper we consider a class of birth–and–death–type models with proliferation, with variable coefficients and under suitable assumptions on the “birth” (amplification) coefficients, the “death” (deamplification) coefficients and the proliferative coefficients (related to the averages of the life-spans) we prove (in Section 3) that the corresponding semigroup is topologically chaotic.

For the death-type model we generalize the result of [3] (in Section 4).

Finally for the “adjoint” (“opposite”) processes we prove that the corresponding semigroups cannot be chaotic (in Section 5).

2. Birth–and–death–type model

We consider the following infinite system of ODEs

ḟ_{n} = (Lf)_{n} ,  n ∈ ℕ_{0} ,  (2.1)

\begin{matrix}(Lf)_{0} = & − a_{0}f_{0} + d_{1}f_{1} , \\ (Lf)_{n} = & − a_{n}f_{n} + b_{n−1}f_{n−1} + d_{n+1}f_{n+1} , n ∈ ℕ , \end{matrix}

where f = {f_{n}(t)}_{n∈ℕ_{0}} is the distribution function, ℕ = {1, 2, …}, ℕ_{0} = ℕ ∪ {0}. The coefficients a_{n} , b_{n} , d_{n} ≥ 0 are given for n ∈ ℕ_{0}; define c_{n} = b_{n} + d_{n} − a_{n}.

Assuming c_{n} ≥ 0 for n ∈ ℕ_{0}, we can interpret Eq. (2.1) as a kinetic equation corresponding to processes characterized by two components: the conservative one (birth–and–death process, formally for d_{0} = c_{0} and c_{n} = 0 ∀ n ∈ ℕ) and the proliferative one(formally for b_{n} = d_{n} = 0 ∀ n ∈ ℕ_{0}). The name of birth–and– death process comes from the population theory, where n is interpreted as the size of a population and the possible events are death and birth of individuals. In this situation the functions f_{n} have the meaning of the probability that the population is of size n at time t. The mathematical theory for the birth–and–death processes can be found in [10], [15] and [18] (see also references therein).

The following very important interpretation of Eq. (2.1) with a_{n} = b_{n}+d_{n}−c_{n}, c_{n} ≥ 0, is possible (see [14], [19] and references therein). Equation (2.1) describes the population of neoplastic cells divided into subpopulations characterized by different levels of cellular resistance to antineoplastic drugs. The subpopulations of cells are labeled by numbers n ∈ ℕ_{0}. The process considered in [14] and [19] is gene stochastic amplification – deamplification process. The cells of n–subpopulation are identified with ξ_{n} copies of the drug resistance gene (ξ_{0} = 0 and ξ_{n} increases with n). Because the number of gene copies can be very large in cancer cells, it is reasonable to use a model with an infinite number of cells subpopulations. The cells of 0–subpopulation have no copies of the gene and are sensitive to the drug. Cells of n–subpopulation are resistant with the increasing level of resistance for increasing n ∈ ℕ. The conservative component of the process describes the mutations of cells: b_{n}Δt + o(Δt), for n ∈ ℕ_{0}, is the chance of one mutation in the n–subpopulation shifting the mutated cell to the n + 1–subpopulation and d_{n}Δt + o(Δt), for n ∈ ℕ, is the chance of one mutation in the n–subpopulation shifting the mutated cell to the n − 1–subpopulation (we assume that d_{0} = 0). The proliferative component of the process is related to the assumption that the moment of death represents the moment of cell division and that the average life-span is given by the coefficient c_{n} for the n–subpopulation (n ∈ ℕ_{0}).

With this interpretation the proper space for the process defined by Eq. (2.1) is l^{1}, where the norm

∥f∥ = ∑_{n=0}^{∞} f_{n} ,  (2.2)

of any element f in the positive cone l_{+}^{1}

l_{+}^{1} = {f ∈ l^{1} ; f_{n} ≥ 0 ∀n ∈ ℕ_{0}} ,  (2.3)

represents the total number of individuals (cells).

Another motivation for studying Eq. (2.1) comes from the theory of kinetic integral equations (generalized Boltzmann models or nonlocal models — see [5] and [16]). In fact, consider the following linear equation

∂_{t}f = Γ ∗ f − ρf ,  (2.4)

in L^{2}(𝕋), where 𝕋 is the 1–dimensional torus, Γ, ρ are given functions on 𝕋 and ∗ is the convolution. The bilinear version of this type of equations was studied in [16]. Equation (2.4) in Fourier transform reads

∂_{t}f_{n} = Γ_{n}f_{n} − ∑_{m=−∞}^{+∞} ρ_{n−m}f_{m} ,  n = 0, ±1, ±2, …,  (2.5)

where f_{n} is the n-th Fourier coefficient of function f, and Eq. (2.5), under the assumption ρ_{n} = 0, ∀ n ≠ −1, 0 leads to Eq. (2.1) with a_{n} = ρ_{0}−Γ_{n} and d_{n} = −ρ_{−1} for n ∈ ℕ_{0}.

Yet another field of applications of Eq. (2.1) is in the extended kinetic theory (cf. [1], [2] and references therein) where this system describes the isotropic scattering of test particles on the medium composed of field particles at various levels of excitation (here two); the scattering consists in the exchange of a quantum of energy between the test and the field particles.

Assuming that a_{n} = a (n ∈ ℕ_{0}) and d_{n} = d (n ∈ ℕ) are independent of n as well as that b_{n} = 0 (∀n ∈ ℕ) and d > a > 0 we obtain the model considered in [17]

ḟ_{n} = (L^{(0)}f)_{n} = −af_{n} + df_{n+1},  n ∈ ℕ_{0} .  (2.6)

The operator L^{(0)} determined by L^{(0)} in l^{1} is bounded so that there exists a semigroup solving Eq. (2.6). It follows that the point spectrum of L^{(0)} is

σ_{P} (L^{(0)}) = {λ = −a + dμ ; |μ| < 1} ,

with the eigenvectors

f(λ) = (1, μ, μ^{2}, μ^{3}, …) ∈ l^{1} .

In [17] it was proved that the corresponding semigroup is chaotic in l^{1} (note that the assumption of Theorem 1.1 are satisfied).

In the paper [3] a generalization of the model introduced in [17] was considered. The generalization consisted in allowing variable coefficients a_{n}, d_{n}, but keeping the assumptions b_{n} = 0 and 0 < a_{n} < d_{n+1} (n ∈ ℕ_{0}). Under the assumption that the sequence {a_{n}}_{n∈ℕ_{0}} tends sufficiently fast to some a > 0 and the sequence {d_{n}}_{n∈ℕ} tends to d > a (as n → ∞) it was proved ([3]) that the corresponding semigroup is topologically chaotic in any l^{p}, 1 ≤ p < ∞, and in c_{0}. It was also proved in [3] that if d_{n} = 0, 0 < a_{n} < b_{n−1} (n ∈ ℕ), the sequence {a_{n}}_{n∈ℕ_{0}} tends sufficiently fast to some a > 0 and the sequence {b_{n}}_{n∈ℕ_{0}} tends to b > a (as n → ∞), then the corresponding semigroup cannot be even hypercyclic in any of the spaces l^{p} (1 ≤ p < ∞), c_{0}.

In the present paper we consider the general equation (2.1) under the assumption that the coefficients a_{n}, b_{n} (for n ∈ ℕ_{0}), d_{n} (for n ∈ ℕ) are nonnegative and

(A1) For some a ∈ [0, ∞[  a_{n} = a + α̃_{n} , ∀n ∈ ℕ_{0} ,  with lim_{n→∞} α̃_{n} = 0 ;

(A2) d_{n} > 0 ∀ n ∈ ℕ ,  ∃ d ∈]0, ∞[ : lim_{n→∞} d_{n} = d ,  inf_{n∈ℕ} ∏_{k=1}^{n} \frac{d_{k}}{d} > 0 ;

(A3) For some b ∈ [0, ∞[,  \overline{lim_{n→∞}}b_{n} = b .

Throughout the paper we use the notation

κ′ = \frac{κ}{d} for each quantity κ,  (2.7a)

and

α_{n} = α̃′_{n} = \frac{α̃_{n}}{d} .  (2.7b)

In Section 3 we show that under Assumption (A1), (A2), (A3), if additionally

(A4) 0 < a + b < d;

and the sequences {α_{n}}_{n∈ℕ_{0}}, {b_{n}d_{n+1}}_{n∈ℕ_{0}} satisfy

(A5) |α_{n}| ≤ q^{n+1} ,  |b′_{n}d′_{n+1}| ≤ q^{2n+4} ,  ∀ n ∈ ℕ_{0}  for some q ∈]0, q_{0}[ ,

where q_{0} ∈]0, 1[ is a critical constant, then the semigroup solving Eq. (2.1) is topologically chaotic.

In Section 4 we consider the death–type model, that is Eq. (2.1) with b_{n} = 0 for all n ∈ ℕ_{0}. In this case the assumption on the critical constant q_{0} can be essentially relaxed. In Section 4 we show that under Assumption (A1) and (A2), if additionally

(A6) 0 < a < d;

and the sequence {α_{n}}_{n∈ℕ_{0}} satisfies

(A7) |α_{n}| ≤ q^{n+1} ,  ∀ n ∈ ℕ_{0} ,  for some q ∈]0, 1[ ,

then the semigroup solving Eq. (2.1), with b_{n} = 0 for all n ∈ ℕ_{0}, is topologically chaotic. The results of Section 4 are a generalization of paper [3].

Finally, in Section 5, we show that the “adjoint” (“opposite”) processes cannot be even hypercyclic.

3. Chaos for the birth–and–death–type model

We denote the infinite matrix of the coefficients of Eq. (2.1) by L. Then by L_{p}, p ∈ {0} ∪ [1, ∞[, we denote bounded operators defined by L in, respectively, c_{0} and l^{p}. The operators L_{p} are bounded, hence they generate uniformly continuous semigroups of operators in, respectively, c_{0} and l^{p}, denoted hereafter by {T_{p}(t)}_{t≥0}.

Firstly we shall focus our attention on L_{1}. The equation for its eigenvectors takes the form

\begin{aligned}λf_{0} & = −a_{0}f_{0} + d_{1}f_{1} , \\ λf_{n} & = −a_{n}f_{n} + b_{n−1}f_{n−1} + d_{n+1}f_{n+1} ,  n ∈ ℕ . \end{aligned}  (3.1)

We introduce the following notation

μ_{n} = \frac{λ + a_{n}}{d_{n+1}} ,  ∀ n ∈ ℕ_{0} .  (3.2)

Clearly we can take f_{0} = 1, then f_{1} = μ_{0} and for n > 1 we obtain the recurrence relation

f_{n+1} = μ_{n}f_{n} − \frac{b_{n−1}}{d_{n+1}} f_{n−1} .  (3.3)

At this point it is worthwhile to note that f_{n} are polynomials in λ. Finding the general form of them seems to be rather difficult and, as it turns out, not necessary since we shall need only some estimates. To this end we note the following lemma.

Lemma 3.1. Assume that there exist constants n_{0} ∈ ℕ, β > 0 and μ̃ > 0 such that

b_{n−1} ≤ βd_{n+1} ,  |μ_{n}| ≤ μ̃ ,  ∀ n ≥ n_{0} .  (3.4)

Then

|f_{n}| ≤ g_{n} ,  (3.5)

for all n ≥ n_{0}, where

g_{n_{0}} = |f_{n_{0}}| ,  g_{n_{0}+1} = |f_{n_{0}+1}| ,  (3.6a)

and

g_{n} = μ̃g_{n−1} + βg_{n−2} ,  n ≥ n_{0} + 2 .  (3.6b)

Proof. Note that (3.5) holds for n = n_{0} and n = n_{0} + 1. Let m ≥ n_{0} + 2 and assume that (3.5) holds for each n ∈ {n_{0}, …, m}. Then

|f_{m+1}| ≤ μ̃|f_{m}| + β|f_{m−1}| ≤ μ̃g_{m} + βg_{m−1} = g_{m+1} ,  (3.7)

and, by induction, (3.5) holds for all n ≥ n_{0}.

Note that some generalizations of Lemma 3.1 are possible by using Perron’s Theorem (see [8]).

The sequence {g_{n}}_{n∈ℕ_{0}} is a Fibonacci sequence and thus its explicit formula can be found by standard calculations. The characteristic equation corresponding to Eq. (3.6b) has the form ω^{2} − μ̃ω − β = 0 and the characteristic roots are given by

ω_{±} = \frac{μ̃ ± \sqrt{μ̃^{2} + 4β}}{2} .  (3.8)

Therefore the general solution to Eq. (3.6b) is

g_{n} = ϱ_{+}ω_{+}^{n} + ϱ_{−}ω_{−}^{n} ,

where ϱ_{+} and ϱ_{−} are constants. Using (3.6a) we obtain the solution to the problem (3.6) as

g_{n} = \frac{1}{ω_{+} − ω_{−}} ((ω_{+}^{n−n_{0}} − ω_{−}^{n−n_{0}})|f_{n_{0}+1}| + (ω_{−}^{n−n_{0}}ω_{+} − ω_{−}ω_{+}^{n−n_{0}})|f_{n_{0}}|) , (3.9)

for all n ≥ n_{0}.

Clearly, if |ω_{±}| ≤ q < 1, then |f_{n}| ≤ |g_{n}| ≤ Mq^{n−n_{0}+1}, n ≥ n_{0}, for some M and f = (f_{0}, f_{1}, …, f_{n}, …) ∈ l^{1}. We have the following lemma.

Lemma 3.2. If Assumptions (A1) – (A4) are satisfied, then there exists r > 0 such that

{λ ∈ ℂ : |λ| < r} ⊂ σ_{P} (L_{1}) .  (3.10)

Proof. By Assumptions (A1) – (A3) we have

μ = lim_{n→∞} μ_{n} = \frac{λ + a}{d} ,

and for any ε > 0 there exists n_{0} such that

|μ_{n}| ≤ |μ| + ε ,  \frac{b_{n−1}}{d_{n+1}} ≤ b′ + ε ,  ∀ n ≥ n_{0} ,  (3.11)

where b′ = \frac{b}{d}. By Lemma 1 and (3.8) we obtain

ω_{±}(λ) = \frac{|λ + a|}{2d} + ε ± ((\frac{|λ + a|}{2d} + ε)^{2} + b′ + ε)^{\frac{1}{2}} .  (3.12)

Consider the following functions

h_{±}(x) = x ± (x^{2} + b′ + ε)^{\frac{1}{2}} ,  x ≥ 0 .  (3.13)

We have |h_{±}(0)| = \sqrt{b′ + ε}, and by Assumption (A4), ε can be chosen such that |h_{±}(0)| < 1. Differentiation gives

h′_{±}(x) = 1 ± \frac{x}{\sqrt{x^{2} + b′ + ε}} > 0 ,  ∀ x > 0 .  (3.14)

Clearly, we have

−1 < −\sqrt{b′ + ε} ≤ h_{−}(x) < 0 ,  ∀ x > 0 .  (3.15)

On the other hand, we have

h_{+}(x_{0}) = 1 ,  for x_{0} = \frac{1 − b′ − ε}{2} ,  (3.16)

and therefore

\sqrt{b′} ≤ h_{+}(x) < h_{+}(x_{0}) = 1 ,  ∀ 0 < x < x_{0} .  (3.17)

Hence, from Assumption (A4), ε > 0 can be chosen small enough so that

a + b < d − 3εd .  (3.18)

From (3.18) it follows that

0 < \frac{a}{2d} + ε < x_{0}  (3.19)

and therefore

|ω_{±}(0)| < 1 .  (3.20)

Since ω_{±} are continuous functions of the complex variable λ, there exists a constant 0 < q < 1 and an open neighbourhood of λ = 0 where |ω_{±}(λ)| ≤ q, thus by the considerations preceding the lemma the eigenvector f belongs to l^{1}.

Some further information on the spectra of operators generated by L with constant coefficients in various weighted l^{1}–spaces can be found in [14] (see also [19]).

Let us denote by f(λ) the eigenvectors corresponding to the eigenvalue λ,

f(λ) = (f_{0}(λ), f_{1}(λ), …) .

Following Theorem 1.1 we take an arbitrary sequence Φ = {Φ_{k}}_{k∈ℕ_{0}} ∈ l^{∞} and consider the function

H_{Φ}(λ) = ∑_{n=0}^{∞} Φ_{n}f_{n}(λ),  (3.21)

where the series is almost uniformly convergent at least in the circle |λ| < r, where r was defined in Lemma 3.2.

In order to use the properties of analytic functions we write H_{Φ} as the power series in μ = \frac{λ+a}{d}. This is possible since, as we noted above, each f_{k} is a polynomial in μ and the series (3.21) is absolutely convergent for |λ| < r, thus the terms can be rearranged in an arbitrary way (see e.g. [13]). Therefore we obtain

H_{Φ}(λ) = ∑_{n=0}^{∞} μ^{n}( ∑_{k=n}^{∞} A_{n,k}Φ_{k}) ,  μ = \frac{λ + a}{d} ,  (3.22)

where A_{n,k} is the coefficient at μ^{n}. It follows immediately from Eq. (3.3) that the matrix {A_{n,k}}_{(n,k)∈ℕ_{0}^{2}} is given by the recurrence relation

A_{n,k} = \{ \begin{matrix}0 & for & k < n , \\ 1 & for & k = n = 0 , \\ ( ∏_{j=1}^{n} d′_{j})^{−1} & for & k = n ≥ 1 , \\ \frac{1}{d′_{k}} (A_{n−1,k−1} + α_{k−1}A_{n,k−1} − b′_{k−2}A_{n,k−2}) & for & k > n , \end{matrix}

where the convention (2.7) is adhered to and A_{−1,k} = 0 for any k.

Since the rearranged series (3.22) is convergent for μ = \frac{r̂+a}{d}, for any r̂ < r, it is convergent for any μ with |μ| < \frac{r̂+a}{d}. Thus, if H_{Φ}(λ) vanishes for |λ| < r, then it vanishes in an open region inside the circle of convergence of the series (3.22) and thus all the coefficients of (3.22) must vanish. Therefore, to show that the last assumption of Theorem 1.1 is satisfied we have to prove that the only bounded solution of the infinite upper–triangular linear system

\begin{aligned}Φ_{0} & + & A_{0,1}Φ_{1} & + & A_{0,2}Φ_{2} & + & … & = 0 , \\ & & A_{1,1}Φ_{1} & + & A_{1,2}Φ_{2} & + & … & = 0 , \\ & & & & A_{2,2}Φ_{2} & + & … & = 0, \\ & & & & & & … & = 0 , \end{aligned}  (3.23)

is the zero solution.

The direct analysis of the behaviour of the coefficients A_{n,k} is difficult mainly due to the appearance of the factor \frac{1}{d′_{n}}. However, it turns out that we can relate A_{n,k} to the analogous coefficients of system with d_{n} = 1. To show this we introduce the auxiliary system

\begin{aligned}λf_{0} & = −α_{0}f_{0} + f_{1} , \\ λf_{n} & = −α_{n}f_{n} + γ_{n−1}f_{n−1} + f_{n+1} ,  n ∈ ℕ , \end{aligned}  (3.24)

where γ_{n} = b′_{n}d′_{n+1} = \frac{b_{n}d_{n+1}}{d^{2}} , for n ∈ ℕ_{0}.

In the same manner as before we find that the power series (3.22) corresponding to the system (3.24) is given by

H_{Φ}(λ) = ∑_{n=0}^{∞} μ^{n}( ∑_{k=n}^{∞} B_{n,k}Φ_{k}) ,  (3.25)

where the matrix {B_{n,k}}_{(n,k)∈ℕ_{0}} is given by the recurrence relation

B_{n,k} = \{ \begin{matrix}0 & for & k < n , \\ 1 & for & k = n , \\ B_{n−1,k−1} + α_{k−1}B_{n,k−1} − γ_{k−2}B_{n,k−2} & for & k > n , \end{matrix}

where B_{−1,k} = 0 for any k.

Lemma 3.3.

A_{n,k} = ( ∏_{j=1}^{k} d′_{j})^{−1}B_{n,k} ,  n, k ∈ ℕ_{0} ,  (3.26)

with the understanding that ∏_{j=1}^{0} = 1.

Proof. By direct calculation we check the validity of (3.26) for all the pairs (n, k) with k ≤ 2 and arbitrary n (note that due to the form of B_{n,k} this reduces to finite numbers if indices n: n ≤ k). Thus we assume that (3.26) is valid for all pairs (n, k̂) where k̂ is an arbitrary number satisfying k̂ < k and n ∈ ℕ_{0}. Let us fix k and take arbitrary n (again there is only the finite range of nontrivial cases: 0 ≤ n ≤ k). We have for k > n

A_{n,k} = \frac{1}{d′_{k}} (A_{n−1,k−1} + α_{k−1}A_{n,k−1} − b′_{k−2}A_{n,k−2})

= \frac{1}{d′_{k}} (\frac{1}{∏_{j=1}^{k−1} d′_{j}} B_{n−1,k−1} + \frac{α_{k−1}}{∏_{j=1}^{k−1} d′_{j}} B_{n,k−1} − \frac{b′_{k−2}}{∏_{j=1}^{k−2} d′_{j}} B_{n,k−2})

= ( ∏_{j=1}^{k} d′_{j})^{−1}(B_{n−1,k−1} + α_{k−1}B_{n,k−1} − b′_{k−2}d′_{k−1}B_{n,k−2})

= ( ∏_{j=1}^{k} d′_{j})^{−1}B_{n,k} ,

and Eq. (3.26) for n ≥ k follows directly. Thus the lemma is valid for all pairs (n, k̂) with arbitrary k̂ ≤ k and n ∈ ℕ_{0} so that we can continue the induction with respect to k which proves Eq. (3.26) for all (n, k) ∈ ℕ_{0}^{2}.

To estimate B_{n,k} we have the following result.

Lemma 3.4. If

|α_{n}| ≤ q^{n+1} ,  and  |γ_{n}| ≤ q^{2n+4} ,  for some q ∈]0, 1[ ,  (3.27)

then

|B_{n,k}| ≤ ∑_{1≤j_{1}≤...≤j_{k−n}≤k} q^{j_{1}+...+j_{k−n}} ,  n < k , (n, k) ∈ ℕ_{0}^{2} .  (3.28)

Proof. We use the same induction argument as in Lemma 3.3. By direct computation we establish the validity of (3.28) for initial values of n and k. Assuming that (3.28) invalid for n = 0 and k̂ < k we obtain

\begin{aligned}|B_{0,k}| & = |α_{k−1}B_{0,k−1} − γ_{k−2}B_{0,k−2}| \\ & ≤ q^{k} ∑_{1≤j_{1}≤...≤j_{k−1}≤k−1} q^{j_{1}+...+j_{k−1}} + q^{2k} ∑_{1≤j_{1}≤...≤j_{k−2}≤k−2} q^{j_{1}+...+j_{k−2}} . \end{aligned}

Multiplication in each term increases the length of each chain of indices to k. Moreover, in the first term the index k will appear only once and in the second will appear only twice, so the sequences of indices will be not repeated. Thus

|B_{0,k}| ≤ ∑_{1≤j_{1}≤...≤j_{k}≤k} q^{j_{1}+...+j_{k}} .

In a similar way we obtain for n > 0

|B_{n,k}| = |B_{n−1,k−1} + α_{k−1}B_{n,k−1} − γ_{k−2}B_{n,k−2}|
≤ ∑_{1≤j_{1}≤...≤j_{k−n}≤k−1} q^{j_{1}+...+j_{k−n}} + q^{k} ∑_{1≤j_{1}≤...≤j_{k−n−1}≤k−1} q^{j_{1}+...+j_{k−n−1}}

+ q^{2k} ∑_{1≤j_{1}≤...≤j_{k−n−2}≤k−2} q^{j_{1}+...+j_{k−n−2}}

≤ ∑_{1≤j_{1}≤...≤j_{k−n}≤k} q^{j_{1}+...+j_{k−n}} .

Let us denote by B_{∞} the operator generated by the matrix {B_{n,k}}_{(n,k)∈ℕ_{0}^{2}} in l^{∞}.

Lemma 3.5. There exists a critical constant q_{0} ∈]0, 1[ such that if Assumption (A5) is satisfied, then the operator B_{∞} is an isomorphism in l^{∞}. Proof. We split B_{∞} as

B_{∞} = I + ( \begin{matrix}0 & B_{0,1} & B_{0,2} & B_{0,3} & … \\ 0 & 0 & B_{1,2} & B_{1,3} & … \\ 0 & 0 & 0 & B_{2,3} & … \\ ⋮ & ⋮ & ⋮ & ⋮ & … \end{matrix} ) = I + B̂_{∞} ,

where I is the identity operator in l^{∞}. We have for any f = (f_{1}, f_{2}, …) ∈ l^{∞}

∥B̂_{∞}f∥_{∞} = sup_{n∈ℕ_{0}} | ∑_{k=n+1}^{∞} B_{n,k}f_{k}| ≤ ∥f∥_{∞} sup_{n∈ℕ_{0}} ∑_{k=n+1}^{∞} |B_{n,k}| .  (3.29)

By Lemma 3.4 we obtain

\begin{aligned}Θ_{n} & = ∑_{k=n+1}^{∞} |B_{n,k}| \\ & ≤ ∑_{1≤j≤n+1} q^{j} + ∑_{1≤j_{1}≤j_{2}≤n+2} q^{j_{1}+j_{2}} + … + ∑_{1≤j_{1}≤...≤j_{k−n}≤k} q^{j_{1}+...+j_{k−n}} + …. \end{aligned}

Each power q^{j} with j > 2 can appear several times in the above sum. However, the number of occurrences cannot exceed the number P (j) of all possible integer solutions to j_{1} + … + j_{r} = j (it is clear that any particular sequence (j_{1}, …, j_{r}) may appear at most once). Thus we can write

Θ_{n} ≤ ∑_{j=1}^{∞} P (j)q^{j} ,

so the estimate is independent of n. Using the result from combinatorics (see e.g. [20], p. 140)

P (j) < \frac{π}{\sqrt{6(j − 1)}} exp (π\sqrt{\frac{2j}{3}} ) ,  for j > 2 ,  (3.30)

we obtain that

\overline{lim_{j→∞}}(q^{j}P (j))^{\frac{1}{j}} ≤ q < 1 .

Thus the Cauchy criterion yields the convergence of each Θ_{n} with the common bound independent of n. Defining

θ(q) = ∑_{j=1}^{∞} P (j)q^{j},

we see that θ is a continuous function of q satisfying θ(0) = 0 so that there exists q_{0} ∈]0, 1[ such that |θ(q)| < 1 for 0 < q ≤ q_{0} and from Eq. (3.29) we get

∥B̂_{∞}f∥_{∞} ≤ θ(q)∥f∥_{∞} < ∥f∥_{∞} .

Using the Neumann series theorem we obtain that B_{∞} is an isomorphism.

Remark 3.1. The critical constant q_{0} in Lemma 3.5 can be chosen not smaller than 0.15. Proof. If ϱ = exp (\frac{π\sqrt{2}}{3}), then for j ≥ 3 we obtain

P (j) < \frac{π}{\sqrt{6(j − 1)}} exp (π\sqrt{\frac{2 j}{3}} ) ≤ \frac{π}{\sqrt{12}} exp (\frac{π\sqrt{2} j}{3} ) = \frac{π}{\sqrt{12}} ϱ^{j} ,

where we used the fact that the function

u(x) = exp (π\sqrt{\frac{2}{3}}(\sqrt{x} − \frac{x}{\sqrt{3}}))

is decreasing for x > 3 and satisfies u(3) = 1. Therefore we obtain

\begin{aligned}θ(q) & ≤ q + 2q^{2} + \frac{π}{\sqrt{12}} ∑_{j=3}^{∞} P (j)q^{j} ≤ q + 2q^{2} + \frac{π}{\sqrt{12}} ∑_{j=3}^{∞}(ϱq)^{j} \\ & = q + 2q^{2} + \frac{π}{\sqrt{12}} \frac{(ϱq)^{3}}{1 − ϱq} . \end{aligned}

We need the range of q > 0 for which

|q + 2q^{2} + \frac{π}{\sqrt{12}} \frac{(ϱq)^{3}}{1 − ϱq} | < 1 .

Numerically we find that the above inequality is satisfied for q ∈ [0 , 0.1514 … [ .

Theorem 3.1. Suppose that Assumptions (A1) – (A5) are satisfied with the critical constant q_{0} as in Lemma 3.5. Then the semigroups {T_{p}(t)}_{t≥0} solving the system (2.1) in l^{p}, 1 ≤ p < ∞, and in c_{0}, are topologically chaotic. Proof. From Lemma 3.2 we see that σ_{P} (L_{1}) contains an open circle with centre at λ = 0, that is σ_{P} (L_{1}) contains open subsets: V_{+} ⊂ ℂ_{+} = {λ : ℜλ > 0}, V_{−} ⊂ ℂ_{−} = {λ : ℜλ < 0}, and an open interval V_{0} of the imaginary axis. We have

\begin{aligned}H_{Φ}(λ) & = ∑_{n=0}^{∞} μ^{n}( ∑_{k=n}^{∞} A_{n,k}Φ_{k}) = ∑_{n=0}^{∞} μ^{n}( ∑_{k=n}^{∞} \frac{Φ_{k}}{∏_{j=1}^{k} d′_{j}} B_{n,k}) \\ & = ∑_{n=0}^{∞} μ^{n}( ∑_{k=n}^{∞} Φ̃_{k}B_{n,k}) . \end{aligned}

Thus if H_{Φ}(λ) vanishes on either set: V_{±} or V_{0}, then the coefficients of the power series must vanish, which is equivalent to B_{∞}Φ̃ = 0. Since we proved in Lemma 3.5 that B_{∞} is an isomorphism on l^{∞}, the only bounded solution is Φ̃ = 0. Hence, if there is a nontrivial bounded Φ for which H_{Φ} vanishes identically on V_{±}, V_{0}, respectively, then by Assumption (A2), Φ̃ would be a nontrivial bounded solution to B_{∞}Φ̃ = 0 which, as we have seen, is impossible. Thus Theorem 1.1 yields that the semigroup {T_{1}(t)}_{t≥0} is topologically chaotic in l^{1}.

Consider the same problem in c_{0} or l^{p} with 1 < p < +∞. In each case the operator L_{p}, p ∈ {0} ∪ [1, ∞[ is bounded and thus generates a uniformly continuous semigroup {T_{p}(t)}_{t≥0}. The eigenvectors belong to each of the above spaces. We have c_{0}^{∗} = l^{1} and (l^{p})^{∗} = l^{r}, with 1/p + 1/r = 1. Hence, if the only solution Φ ∈ l^{r}, 1 ≤ r ≤ +∞, of the equation B_{r}Φ = 0 (where B_{r} is the operator generated in l^{r} by the matrix {B_{n,k}}_{(n,k)∈ℕ_{0}^{2}}) is the zero solution, then the semigroup {T_{p}(t)}_{t≥0} is topologically chaotic in each of these spaces. But any element of l^{r} is necessarily bounded, thus the statement follows from the unique solvability of B_{∞}Φ = 0 in l^{∞}.

4. Chaos for the death–type model

In this section we consider the death–type model, that is, Eq. (2.1) with b_{n} = 0 for n ∈ ℕ_{0}. In this case the estimates leading to Theorem 3.1 can be substantially improved. This is possible as the transpose to the matrix that corresponds here to the matrix {A_{n,k}}_{(n,k)∈ℕ_{0}} can be explicitly inverted. We consider the infinite system of ODEs in l^{1}

ḟ_{n} = −a_{n}f_{n} + d_{n+1}f_{n+1} ,  n ∈ ℕ_{0} ,  (4.1)

where the coefficients satisfy Assumptions (A1), (A2), (A6) and (A7).

Let us denote by K the matrix of coefficients of (4.1) and by K_{p}, p ∈ {0}∪[1, ∞[, the operator determined by K in the spaces c_{0} and l^{p}, respectively. These operators are bounded and therefore generate semigroups of uniformly bounded operators {S_{p}(t)}_{t≥0} in the respective spaces.

As before, firstly we shall focus our attention on K_{1}. The equation for eigenvectors reads

λf_{n} = −a_{n}f_{n} + d_{n+1}f_{n+1} ,  n ∈ ℕ_{0} .  (4.2)

Formally we obtain its solution as

f(λ) = (f_{n}(λ))_{n∈ℕ_{0}},

where f_{0}(λ) = 1, and

f_{n}(λ) = ∏_{j=0}^{n−1} \frac{λ + a_{j}}{d_{j+1}} , n ∈ ℕ .

It is clear that the assumptions of Lemma 3.2 are satisfied and the point spectrum of K_{1} has the structure required by Theorem 1.1.

Following this theorem, as in Section 3, we take an arbitrary sequence Φ̃ ∈ l^{∞} and consider the function

FΦ̃(λ) = ∑_{n=0}^{∞} Φ̃_{n}f_{n}(λ) ,  (4.3)

where the series is uniformly convergent in some open set of the complex plane containing zero. Due to the simpler structure of the problem we can simplify the above expression at this moment by writing

F_{Φ}(μ) = FΦ̃(λ) = ∑_{n=0}^{∞} \frac{Φ̃_{n}}{∏_{k=1}^{n} d′_{k}} ∏_{j=0}^{n−1} (\frac{λ + a}{d} + α_{j}) = ∑_{n=0}^{∞} Φ_{n} ∏_{j=0}^{n−1}(μ + α_{j}) ,

where μ = \frac{λ+a}{d}, and where we used the notation (2.7) as well as the convention

that ∏_{j=0}^{−1} = 1. The function F_{Φ} is analytic in an open neighbourhood of μ = 0.

By Assumption (A2), if Φ̃ ∈ l^{∞}, then also Φ ∈ l^{∞}. Hence, showing that F_{Φ} ≡ 0 yields Φ = 0 for any Φ ∈ l^{∞} proves the same statement for FΦ̃. Consequently we concentrate on F_{Φ}.

In order to use the properties of analytic functions we write F_{Φ}(μ) as a power series in μ by multiplying out the brackets and rearranging the terms. We obtain

F_{Φ}(μ) = ∑_{n=0}^{∞} μ^{n}( ∑_{k=n}^{∞} C_{n,k}Φ_{n}) ,  (4.4)

where

C_{n,k} = \{ \begin{matrix}0 & for & k < n , \\ 1 & for & k = n , \\ ∑_{0≤j_{1}<...<j_{k−n}≤k−1} α_{j_{1}} · . . . · α_{j_{k−n}} & for & k > n . \end{matrix}

As in Section 3, in order to use Theorem 1.1 we have to show that the only l^{∞}–solution of the infinite upper-triangular linear system

\begin{aligned}Φ_{0} & + & C_{0,1}Φ_{1} & + & C_{0,2}Φ_{2} & + & … & = 0 , \\ & & Φ_{1} & + & C_{1,2}Φ_{2} & + & … & = 0 , \\ & & & & Φ_{2} & + & … & = 0, \\ & & & & & & … & = 0 , \end{aligned}  (4.5)

is the zero solution.

Let C_{∞} be the operator generated by the matrix {C_{n,k}}_{(n,k)∈ℕ_{0}} in l^{∞}, let C_{1}^{T} be the operator generated by the transposed matrix in l^{1} and finally let C_{0} be the restriction of C_{∞} to the space c_{0}. We have

Lemma 4.1. Let Assumption (A1), (A2) and (A7) be satisfied. Then (i) C_{∞} is a bounded operator in l^{∞}, (ii) C_{1}^{T} is a bounded operator in l^{1}, (iii) (C_{1}^{T})^{∗} = C_{∞}, (iv) C_{0} is bounded in c_{0} and C_{0}^{∗} = C_{1}^{T}. Proof. (i) The estimates are similar to that of Lemma 3.5. Let f = (f_{1}, f_{2}, …) ∈

l^{∞}, then C_{∞}f = g = (g_{1}, g_{2}, …), where g_{n} = ∑_{k=n}^{∞} C_{n,k}f_{k} and

∥C_{∞}f∥_{∞} = sup_{n∈ℕ_{0}} | ∑_{j=n}^{∞} C_{n,k}f_{k}| ≤ ∥f∥_{∞} sup_{n∈ℕ_{0}} ∑_{j=n}^{∞} |C_{n,k}| .

The sum takes the form

\begin{aligned}Θ_{n} & = 1 + | ∑_{0≤j≤n} α_{j}| \\ & + | ∑_{0≤j_{1}<j_{2}≤n+1} α_{j_{1}}α_{j_{2}} | + … + | ∑_{0≤j_{1}<...<j_{k−n}≤k−1} α_{j_{1}} · . . . · α_{j_{k−n}} | + …. \end{aligned}

It is clear that we can assume that for n < n_{0} we have |α_{n}| ≤ Mq^{n+1} for some constant M > 0. With this we get the estimate

\begin{aligned}Θ_{n} & ≤ M^{n_{0}}(1 + ∑_{1≤j≤n+1} q^{j} \\ & + ∑_{1≤j_{1}<j_{2}≤n+2} q^{j_{1}+j_{2}} + … ∑_{1≤j_{1}<...<j_{k−n}≤k} q^{j_{1}+...+j_{k−n}} + …) . \end{aligned}

Then, as in Lemma 3.5, we can use the partition function P = P (j) to get

Θ_{n} ≤ M^{n_{0}} ∑_{j=0}^{∞} P (j)q^{j} < +∞ ,

provided that |q| < 1. These estimates are independent of n which gives the boundedness of C_{∞}.

(ii) The argument is similar. Let f ∈ l^{1}, then C_{1}^{T} f = g, where g_{n} = ∑_{k=0}^{n} C_{k,n}f_{k} and

∥C_{1}^{T} f∥_{1} = ∑_{n=0}^{∞} | ∑_{k=0}^{n} C_{k,n}f_{k}| ≤ ∑_{k=0}^{∞} f_{k} ∑_{n=k}^{∞} |C_{k,n}| = ∑_{k=0}^{∞} f_{k}Θ_{k} ≤ const∥f∥_{1} .

(iii) Since C_{1}^{T} is bounded and the elements e_{n} = (0, 0, …, 0, 1, 0, …) are linearly dense in l^{1}, it is enough to show that

⟨C_{1}^{T} e_{n} , f⟩ = ⟨e_{n} , C_{∞}f⟩ ,

for any f ∈ l^{∞}. But C_{1}^{T} e_{n} = (0, …, 0, 1, C_{n,n+1}, …), and

⟨C_{1}^{T} e_{n} , f⟩ = ∑_{k=n}^{∞} C_{n,k}f_{k} .

On the other hand, C_{∞}f = { ∑_{k=n}^{∞} C_{n,k}f_{k}}_{n=0}^{∞}, thus

⟨e_{n} , C_{∞}f⟩ = ∑_{k=n}^{∞} C_{n,k}f_{k}

and (iii) follows. (iv) To prove that C_{0} is a bounded operator in c_{0}, it is enough to show that c_{0} is an invariant subspace for C_{∞}. Using more carefully the results of (i) we obtain

|g_{n}| = | ∑_{k=n}^{∞} C_{n,k}f_{k}| ≤ sup_{k≥n} |f_{k}| ∑_{k=n}^{∞} |C_{n,k}| ≤ Θ_{n} sup_{k≥n} |f_{k}| .

Since we proved that Θ_{n} are bounded uniformly with respect to n, we see that if f ∈ c_{0}, then also g ∈ c_{0}.

The statement C_{0}^{∗} = C_{1}^{T} follows from c_{0}^{∗} = l^{1} and calculations similar to (iii).

Consider the transpose matrix C^{T} = {C_{k,n}^{T}}_{(k,n)∈ℕ_{0}^{2}}, that is C_{k,n}^{T} = C_{n,k}, (k, n) ∈ ℕ_{0}^{2}. Define the matrix D = {D_{k,n}}_{(k,n)∈ℕ_{0}^{2}} such that D_{k,n} = (−1)^{k−n}C_{k,n}^{T} for all (k, n) ∈ ℕ_{0}^{2}.

Lemma 4.2. The infinite matrix D is a right-inverse of C. Proof. We prove that

∑_{j=n}^{k} C_{j,k}D_{j,n} = δ_{kn} ,  ∀ (k, n) ∈ ℕ_{0}^{2} ,  n ≤ k ,  (4.6)

where δ_{kn} is the Kronecker delta. From the definition it follows that (4.6) is satisfied for k ≤ n, thus we have only to prove that

∑_{j=n}^{k} C_{j,k}D_{j,n} = 0 ,  ∀ n < k .  (4.7)

As the proof is not particularly intuitive, let us indicate the main ideas behind the considerations that follow below. Namely, it turns out that C_{n+1,k}D_{n+1,n} contains all the terms appearing in C_{n,k}D_{n,n} with the opposite signs so that adding the two first summands of Eq. (4.7) we wipe out all the terms coming from the first summand and this continues as we increase j. Unfortunately, at some levels the terms that had cancelled each other before could appear again. Thus the procedure presented below is aimed at partitioning the set of all the terms that appear in Eq. (4.7) into disjoint classes containing the terms that differ only in sign and proving that they cancel each other within each class.

Fix k and n, n < k. Let Υ_{m} the set of indices over which the summation is carried over in the term j = n+m above. This set is the tensor product of sets of all sequences J_{m} : 0 ≤ j_{1} < j_{2} < … < j_{k−(n+m)} ≤ k − 1 and I_{m} : 0 ≤ i_{1} ≤ … i_{m} ≤ j. If we count each term according to its multiplicity in this set, then for any m the sequence of indices will have k−n elements. We write any s ∈ Υ_{m} as a nondecreasing sequence and for s = (s_{1}, …, s_{k−n}) ∈ Υ_{m} we symbolically denote

ᾱ_{s} = ∏_{i=1}^{k−n} α_{s_{i}}

so that

(−1)^{j−n}C_{j,k}D_{j,n} = (−1)^{m}C_{n+m,k}D_{n+m,n} = ∑_{s∈Υ_{m}} ᾱ_{s} .

For further use we also define

Υ = ⋃_{m=0}^{k−n} Υ_{m}

(the sum is not disjoint). Thus we have to prove

∑_{Υ_{m}⊂Υ}(−1)^{m} ∑_{s∈Υ_{m}} ᾱ_{s} = 0 .  (4.8)

Let us first observe that in any sequence J_{m} taking the tensor product with I_{m} does not change the ”tail” consisting of all the indices which are larger than j, as the indices in I_{m} can equal at most j. We shall then separate in Υ_{m} these sequences which contain the longest possible (at m level) subsequence of indices that are larger that j (such a subsequence is necessarily strictly increasing as it can come only from J_{m}). It follows that such a subsequence is given by j_{1} < … < j_{k−n+m}, j_{1} > j and it is clear that no sequence s containing such a tail can appear for any m̂ > m (the reason for this is that for m̂ > m the strictly increasing sequences taken from Jm̂ are shorter than k − n + m and they are supplemented by elements from Im̂, which are at most equal to j).

Hence for each m we select the sequences s ∈ Υ_{m} of the form

s_{I_{m}} ⊕ s_{J_{m}} = (i_{1}, …, i_{m}, j_{1}, …, j_{k−(n+m)}) ,  (4.9)

where s_{I_{m}} = (i_{1}, …, i_{m}) ∈ I_{m} and s_{J_{m}} = (j_{1}, …, j_{k−(n+m)}) ∈ J_{m} so that i_{m} ≤ j < j_{1}. Note that in particular s_{I_{m}} is empty for m = 0 and s_{J_{m}} is empty for n + m = k. In such a case we have only either s_{J_{m}} or s_{I_{m}}, respectively.

Let the sequence s_{I_{m}} contain a strictly increasing subsequence j_{s_{1}} < j_{s_{2}} < … < j_{s_{r}} for some r ≤ m. We find all possible sequences sm̂ with m̂ < m which give the sequence (4.9). The smallest m̂ for which such a sequence can appear is when all the strictly increasing indices are contributed from Jm̂

(j_{s_{1}}, …, j_{s_{r}}, j_{1}, …, j_{k−(n+m)}) ∈ Jm̂ ,  (4.10)

that is for m̂ = m − r; then we will have only one sequence of this form. Next, at m̂ = m − r + 1 the contributions will come from sequences of the form(4.10) with one of the indices s_{i} missing. Since again for a given missing index s_{i} there is only one sequence in Υm̂ producing (4.9), we see that (4.9) can be obtained by r sequences in Υm̂. Continuing, for m̂ = m − r + l the contribution comes from the sequences of the form (4.10) with any combination of l indices missing, that is

(4.9) can be obtained by ( \begin{matrix}r \\ l \end{matrix} ) sequences from Υ_{m−r+l}. Note moreover that the

contribution from Υ_{m−r} appeared with factor (−1)^{m−r} and the subsequent ones, for given l, are multiplied by (−1)^{m−r+l}.

Let us denote the subset of Υ containing all the sequences (4.9) by Υ_{I_{m}J_{m},r}, where r denotes the number of different indices in I_{m}. By the considerations above, the sets Υ_{I_{m}J_{m},r} constitute a partition of Υ into disjoint sets and moreover for all s ∈ Υ_{I_{m}J_{m},r} we have ᾱ_{s} = α_{I_{m}J_{m},r} (independent of s). Thus we have

∑_{Υ_{m}⊂Υ}(−1)^{m} ∑_{s∈Υ_{m}} ᾱ_{s} = ∑_{\{(I_{m}J_{m},r); Υ_{ImJm,r}⊂Υ\}}(−1)^{m−r}α_{I_{m}J_{m},r} ∑_{l=0}^{r}(−1)^{l} ( \begin{matrix}r \\ l \end{matrix} )

= ∑_{{(I_{m}J_{m},r); Υ_{ImJm,r}⊂Υ}}(−1)^{m−r}α_{I_{m}J_{m},r}(1 − 1)^{r} = 0 ,

so that (4.8) is proved.

Lemma 4.3. Let Assumptions (A1), (A2) and (A7) be satisfied. Then the operator D_{1} generated by the matrix D in l^{1} is bounded and it is the inverse of C_{1}^{T}. Moreover C_{∞} is an isomorphism on l^{∞}. Proof. The norm estimates of D_{1} in l^{1} are analogous to the estimates of C_{1}^{T} so that D is bounded. From Lemma 4.2 we see that D_{1} is the right-inverse of C_{1}^{T} and since the latter is clearly injective, D_{1} is the inverse, thus C_{1}^{T} is an isomorphism in l^{1}. By Lemma 4.1 (iii) we obtain that C_{∞} is an isomorphism in l^{∞}.

We now formulate a theorem the proof of which is the same as of Theorem 3.1.

Theorem 4.1. Suppose that Assumptions (A1), (A2), (A6) and (A7) are satisfied. Then the semigroups {S_{p}(t)}_{t≥0} solving the system (4.1) in l^{p}, 1 ≤ p < ∞, and in c_{0} are topologically chaotic.

5. “Adjoint” models

We consider the “adjoint” (“opposite”) process to the one considered in Section 3, that is, one with small death rate and possibly large birth-rate. The system of governing equations for such a process can be obtained as the transpose of (2.1)

\begin{aligned}ḟ_{0} & = −a_{0}f_{0} + b_{0}f_{1} , \\ ḟ_{n} & = −a_{n}f_{n} + d_{n}f_{n−1} + b_{n}f_{n+1} ,  n ∈ ℕ , \end{aligned}  (5.1)

We adopt the same assumptions as in Section 3, that is, (A1) – (A5) are supposed to be satisfied. Then it is clear that the matrix L^{T} defined by the rhs of Eq. (5.1) defines bounded operators L_{0}^{T} in c_{0}, and L_{p}^{T} in l^{p}, 1 ≤ p < ∞, respectively. As we noted in Section 3, the transpose matrix L defines a bounded operator L_{1} in l^{1} and L_{r} in l^{r}, where \frac{1}{p} + \frac{1}{r} = 1, with (L_{0}^{T})^{∗} = L_{1} and (L_{p}^{T})^{∗} = L_{r}. Denote the semigroup generated by L_{p}^{T}, p ∈ {0}∪[1, ∞[ , in the respective space, by {𝔗_{p}(t)}_{t≥0}. Let us also recall that in a separable Banach space a semigroup is chaotic if it is hypercyclic (topologically transitive) and has a dense set of periodic points. Then we have

Theorem 5.1. Suppose that Assumptions (A1) – (A5) are satisfied. Then the semigroups {𝔗_{p}(t)}_{t≥0} solving the system (5.1) are not hypercyclic for any p ∈ {0} ∪ [1, ∞[. Proof. Let p ∈ {0} ∪ [1, ∞[. It follows that the eigenvectors defined by (3.3) are by Lemma 3.2 in any l^{r}, 1 ≤ r < ∞, so in particular they are eigenvectors of any (L_{p}^{T})^{∗}, which implies that the point spectrum σ_{P} ((L_{p}^{T})^{∗}) of (L_{p}^{T})^{∗} is not empty. This, by Theorem 1.2 (ii), shows that L_{p}^{T} cannot generate a hypercyclic semigroup.

Similar arguments apply to the “adjoint” process (a birth–type model) corresponding to the one defined by (4.1)

\begin{aligned}ḟ_{0} & = −a_{0}f_{0} , \\ ḟ_{n} & = −a_{n}f_{n} + d_{n}f_{n−1} ,  n ∈ ℕ , \end{aligned}  (5.2)

with the matrix of the rhs obtained as the transpose of K. Let {𝔖_{p}(t)}_{t≥0}, p ∈ {0} ∪ [1, ∞[ be the corresponding semigroup in c_{0} and l^{p}, respectively.

Theorem 5.2. Suppose that Assumptions (A1), (A2), (A6) and (A7) are satisfied. Then the semigroups {𝔖_{p}(t)}_{t≥0} solving the system (5.2) are not hypercyclic for any p ∈ {0} ∪ [1, ∞[.

Acknowledgment

The research of the first author (J.B.) was partially supported by National Research Foundation of South Africa. The research was initiated during the visit of the second author (M.L.) at the University of Natal in Durban, which was supported by the National Research Foundation of South Africa and the University of Natal Research Fund. The work of M.L. was done within the framework of EC Programme “Improving the Human Research Potential and Socio–Economic Knowledge Base” —Research Training Networks, Con. N. HPRN–CT–2000–00105, “Using Mathematical Modelling and Computer Simulation to Improve Cancer Therapy”.

Bibliography

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