Large time behavior of the logistic age-structured population model in a changing environment

Abstract

Population growth is governed by many external and internal factors. In order to study their effects on population dynamics, we develop an age-structured time-dependent population model with logistic-type nonlinearity. We prove existence of a unique nonnegative bounded solution. Our main concern is to study asymptotic behavior of a solution in the general case, and especially for a periodic environment. We use the method of lower and upper solutions known in the theory of integral equations to formulate lower and upper boundaries of population density. In the periodic case, we discover a connection between the period of oscillation and its effect on population growth.

Keywords

Age-structure time-variability density-dependency asymptotic behavior of solution

1. Introduction

In the world that is rapidly changing, preservation of diversity in nature, prevention of eventual extinctions or explosions of species, or even land-use management depend on our understanding of the mechanisms that rule population growth. It has been observed that natural environments usually show temporal variation. This may be caused by natural or biological factors, for example a change of seasons, daily variation in temperature and light, exposure to diseases or predation, or by human activities which lead to climate changes, pollution or reduction of habitat’s quality and size. While organisms can adapt to the environmental variations up to some level, a combination of natural and human-caused changes can be too large for populations to cope with; see for instance [2,26]. The detrimental effects for population growth can be seen in decreased abundance of many species and loss of biodiversity; see e.g., [3,28]. Despite the fact that the birth and death rates, competition, or predation are affected by temporal changes, the majority of mathematical models of population dynamics do not include explicitly time-dependent model parameters.

Namely, we usually consider that individuals in a population differ with respect to their age, size, or some other characteristic. In order to reproduce, organisms must be of a certain age, size, or stage. This puts the population structure among the most significant factors for population growth. By dividing a population into classes of individuals of the same age and assuming that individuals in each age-class are identical, the linear age-structured models are developed by Sharpe and Lotka [27], McKendrick [21] and von Foerster [30]. For general structured population models, we refer to Diekmann et al. [11] and Iannelli [14].

Another important observation is that the amount of available resources is limited by the environment, which makes individuals in a population compete for them. This is included in population models by introducing the carrying capacity and density-dependent mortality, as in the Verhulst logistic model of population growth; see for example [14]. The carrying capacity represents the maximal population that the environment can support and the density-dependent mortality is additional mortality due to the intraspecific competition. In the case of an age-structured population, the density-dependent models are studied by, e.g., Gurtin [13], Marcati [19], Webb [33] and [32] and Chan and Guo [4]. These models partially complement the above-mentioned linear age-structured models and include the vital rates that explicitly depend on the population density, but not on time. Price in [22] argues that the carrying capacity may depend on a number of parameters, including age, time, population density etc. Fairly general models with respect to the density-dependence are considered by Prüss [23] and Diekmann et al. [11]. However, these models do not include time-dependent vital rates.

When it comes to the temporal variability, it is included in the age-structured models by Chipot [5] and Elderkin [12] through the birth and death rates that explicitly depend on age, time and the total population. However, the large-time behavior of a solution is not studied for these models. Natural environments are usually changing regularly and nearly periodically. To describe the influence of these fluctuations on organisms and as a mathematical convenience, it is often assumed that the model parameters are periodic functions. The classical logistic model with periodic carrying capacity has been studied by Coleman [6]. Periodicity is often combined with the assumption that the amplitude of fluctuations is small. This allows application of classical perturbation techniques for analysis of the large-time behavior of the solution. For the unstructured models with periodic parameters we refer readers to [8], while for the age-structured models there are works by Cushing [9] and [10] and Kozlov et al. [17]. This is somewhat restrictive approach, since it precludes the study of large variations in the model parameters. For a more general treatment of periodicity in the age-structured models, we refer to [16].

From the previous considerations, it follows that three important factors for population growth, the population structure, density dependence and temporal variability are rarely studied together. It is natural to assume that their effects are intertwined and that may give rise to a complex population dynamics. Thus, the purpose of this paper is to formulate an age-structured population model which includes the intraspecific competition and explicitly time-dependent vital rates. Especially important for investigation of permanence is to study the large-time behavior of a solution. For this, we will use the time-dependent upper and lower boundaries of a solution, similar to the approach in [17].

In order to model density-dependence, we would like to point out that individuals from different age-classes might not compete for the same resource and that competition strength may vary with respect to age and time. This is especially noticeable in the case of the stage-structured populations, such as insect populations. In this case, individuals in different stages of the life cycle may have quite different morphology and therefore different needs. The vital rates that depend on the total population, which suggests that every individual is competing with every other individual in the population, might not be the correct assumption. The population-level competition can be seen as an extreme case of competition, while on the other side of the spectrum would be the situation when only individuals of the exactly same age are competing.

In the general settings, the death rate is given by $\begin{matrix} μ (a, t, ν (t)), where ν (t) = \int_{0}^{\infty} α (a, t) n (a, t) d a, \end{matrix}$ and $α (a, t)$ is the weight function. In the special case for $α (a, t) \equiv 1$ , the weighted integral $ν (t)$ reduces to the total population $N (t)$ , the mortality term is the same as in [5] or [13] and it reflects competition on the population level. On the other hand, the mortality $μ (a, t, n (a, t))$ denotes another specific case when only individuals of the same age compete.

In what follows, we assume that competition exists only within each age-class and that density-dependent mortality takes the specific form $μ (a, t, n (a, t))$ . We take this assumption as a mathematical convenience, which allows us to find an explicit solution and study its asymptotics in detail. The most notable difference between the linear model and the density-dependent models is the exponential growth in the former case and bounded growth in the latter case. As we will see, the model with $μ (a, t, n (a, t))$ has a nonnegative bounded solution, which implies that even this very restricted competition can make qualitative difference in behavior of a solution. Hence the model can be seen as a starting point in the analysis of more realistic cases which would involve the function $μ (a, t, ν (t))$ .

To set up the model, we begin with the balance equation $\begin{array}{l} (1) & \frac{\partial n (a, t)}{\partial t} + \frac{\partial n (a, t)}{\partial a} & = - μ (a, t) n (a, t) (1 + \frac{n (a, t)}{L (a, t)}), a, t > 0, \end{array}$ where $n (a, t)$ is the number of individuals of age a at time t, $μ (a, t)$ is per capita death rate and $L (a, t)$ is the regulating function, which represents limitations imposed on individuals by the environment (or available resource per capita). If the resources are sparse, the density-dependent mortality increases, i.e., form $L (a, t) \to 0$ it follows that $\frac{μ (a, t) n^{2} (a, t)}{L (a, t)} \to \infty$ . Conversely, if the resources are abundant, overcrowding effects are milder, the density-dependent mortality decreases and the logistic model converges to the linear age-structured model, i.e., $L (a, t) \to \infty$ implies $\frac{μ (a, t) n^{2} (a, t)}{L (a, t)} \to 0$ .

For simplicity, we assume that the birth rate is unaffected by the population density. The boundary and the initial conditions are $\begin{array}{l} (2) & n (0, t) & = \int_{0}^{\infty} m (a, t) n (a, t) d a, t > 0, \end{array}$ and $\begin{array}{l} (3) & n (a, 0) & = f (a), a > 0, \end{array}$ where $m (a, t)$ is per capita birth rate and $f (a)$ is the initial distribution of population into age-classes.

Finding a solution to the model (1)–(3) and proving its uniqueness can be reduced to finding a fixed point to the equation $\begin{matrix} ρ (t) = K ρ (t) + F f (t), \end{matrix}$ where $ρ (t)$ denotes the number of newborns and $K$ and $F$ are certain integral operators. To study the asymptotic behavior of $ρ (t)$ , we use the upper and lower solutions of the previous integral equation.

If the model parameters are time-independent functions, we formally assume that ρ is a constant function. Then, for large t, we obtain the characteristic equation $ρ = K ρ$ . A solution to this equation is positive, provided that the net reproductive rate is greater than one. It can be used to define the exact upper and lower bounds for the number of newborns and for the total population.

If the model parameters are time-dependent functions, we construct two auxiliary time-independent problems as follows. The best case scenario is defined by taking the minimal death rate, the maximal birth rate and abundant resources, while for the worst case scenario the opposite is true: the maximal death rate, the minimal birth rate and sparse resources. The vital rates and the regulating function are time-independent in these scenarios, since they are obtained by taking the supremum or infimum with respect to time of the functions in the original time-dependent problem. The best case scenario generates upper bounds for population growth and the worst case scenario gives lower bounds. In all other cases, the number of newborns and the total population fluctuate between boundaries provided by these two extremes. Thus, it is possible to have a prognosis for population growth in time-dependent case by studying behavior of upper and lower bounds in the time-independent cases.

It is clear that fluctuations of the environment or changes in the vital rates are quite often recurring. Some of those variations can be regular, such as seasonal changes of temperature, humidity or sunlight. The others are periodical, but perhaps not as predictable. Examples of those are exposure to harmful substances, presence of predators, amount of available food and so on. This is not a novelty in the existing literature. For example, Boyce and Daley in [2], Coleman in [6], Cook in [7], Cushing in [8] and [9], Roughgarden in [25], Tuljapurkar in [29] considered oscillatory or stochastic environments for different population models. Following this line of thought, we refine our results for periodically varying environments and find the explicit forms of upper and lower solutions and their average values.

A relation between the frequency of oscillation and its influence on population growth has been noted in the linear discrete-time case by Tuljapurkar [29] and in the linear age-structured continuous-time case by Kozlov et al. [17]. May [20] argues that the population abundance and the eventual extinction are related to the responsiveness of the species and to the temporal pattern of the carrying capacity. Levins [18] and May [20] claim that oscillations keep population size below its carrying capacity. We analyze the logistic model (1)–(3) and hypothesize that certain oscillations may be detrimental for population growth, while others are beneficial. Whether the average density of newborns and the average population density are decreased or increased by the oscillations depends on the life-history and on the frequency of environmental changes.

2. Age-structured logistic model in a variable environment

The population model (1)–(3) allows us to study dynamics of an age-structured population in the variable environment under the assumption that competition occurs within each age-class and its strength changes in time. For the biological purposes, the functions describing vital rates and competition between individuals must fulfill certain requirements. The assumptions are as follows.

The constant $A_{μ}$ is the maximal length of life of individuals in population and the interval of fertility is $(a_{m}, A_{m})$ . Notice that $(a_{m}, A_{m}) \subset [0, A_{μ}]$ .

The vital rates $m (a, t)$ and $μ (a, t)$ , the initial distribution $f (a)$ and the regulating function $L (a, t)$ are measurable, nonnegative functions such that:

m is bounded for all $a, t ⩾ 0$ , $\begin{matrix} \begin{matrix} m (a, t) ⩾ δ > 0 if a \in (a_{1}, a_{2}), where 0 < a_{m} < a_{1} < a_{2} < A_{m}, \\ m (a, t) = 0 if a \notin (a_{m}, A_{m}) . \end{matrix} \end{matrix}$

$0 < c_{μ} ⩽ μ (a, t) ⩽ C_{μ} < \infty$ for $a \in (0, A_{m})$ , $\begin{matrix} \int_{T}^{A_{μ} + T} μ (a, t) d a = \infty for T ⩾ 0 . \end{matrix}$

f is bounded for $a ⩾ 0$ , $\begin{array}{l} f (a) ⩾ δ_{1} > 0 for a \in (b_{1}, b_{2}), where b_{2} < a_{2}, \\ f (a) = 0 for a > A_{μ} . \end{array}$

$0 < L_{1} ⩽ L (a, t) ⩽ L_{2} < \infty$ for all $a, t ⩾ 0$ .

The following notation will be used throughout the text: $\begin{array}{l} (4) & Q (a, t) = m (a, t) e^{- \int_{0}^{a} μ (v, v + t - a) d v}, t > a, \\ (5) & π (a, t) = \int_{0}^{a} \frac{μ (v, v + t - a) e^{- \int_{0}^{v} μ (s, s + t - a) d s}}{L (v, v + t - a)} d v, t > a, \\ (6) & ϕ (a, t) = \int_{a - t}^{a} \frac{μ (v, v + t - a) e^{- \int_{a - t}^{v} μ (s, s + t - v) d s}}{L (v, v + t - a)} d v, a > t, \\ (7) & ψ (a, t) = m (a, t) e^{- \int_{a - t}^{a} μ (v, v + t - a) d v}, a > t . \end{array}$

According to assumptions (ii) and (iv), from (5) follows that $\begin{array}{l} \frac{1}{L_{2}} (1 - e^{- \int_{0}^{a} μ (s, s + t - a) d s}) ⩽ π (a, t) ⩽ \frac{1}{L_{1}} (1 - e^{- \int_{0}^{a} μ (s, s + t - a) d s}), \end{array}$ which implies in particular $\begin{array}{l} (8) & 0 < δ_{0} ⩽ π (a, t) for a \in (a_{m}, A_{m}), t ⩾ 0 \end{array}$ where $δ_{0}$ does not depend on a and t. Similarly, from (6) we get $\begin{array}{l} (9) & \frac{1}{L_{2}} (1 - e^{- \int_{a - t}^{a} μ (s, s + t - a) d s}) ⩽ ϕ (a, t) ⩽ \frac{1}{L_{1}} (1 - e^{- \int_{a - t}^{a} μ (s, s + t - a) d s}) . \end{array}$

Our first result shows that if the function n is a solution to the problem (1)–(3), then it satisfies a certain integral equation.

Theorem 2.1.
Let $n \in C^{1} ((0, \infty) \times (0, \infty))$ be a solution to the population problem ( 1 )–( 3 ). Then the function $ρ (t)$ defined by $\begin{array}{l} ρ (t) = n (0, t), t ⩾ 0, \end{array}$ satisfies the integral equation $\begin{array}{l} (10) & ρ (t) & = \int_{0}^{t} \frac{Q (a, t) ρ (t - a)}{1 + ρ (t - a) π (a, t)} d a + \int_{t}^{\infty} \frac{ψ (a, t) f (a - t)}{1 + f (a - t) ϕ (a, t)} d a, \end{array}$ and for other value of a, $n (a, t)$ can be found by $\begin{matrix} (11) & n (a, t) = \{\begin{matrix} \frac{ρ (t - a) e^{- \int_{0}^{a} μ (v, v + t - a) d v}}{1 + ρ (t - a) π (a, t)}, & a < t \\ \frac{f (a - t) e^{- \int_{a - t}^{a} μ (v, v + t - a) d v}}{1 + f (a - t) ϕ (a, t)}, & a ⩾ t . \end{matrix} \end{matrix}$

The proof of the theorem is in the Appendix. Its importance is in the fact that it allows us to introduce a generalized solution ρ to the problem (1)–(3), where ρ is a bounded function given by (10). In order to show that the problem (1)–(3) has a unique solution, we will prove that solution to (10) is unique. We begin by writing equation (10) in the operator form: $\begin{matrix} (12) & ρ (t) = K ρ (t) + F f (t), t ⩾ 0, \end{matrix}$ where $\begin{matrix} (13) & K ρ (t) = \int_{0}^{t} \frac{Q (a, t) ρ (t - a)}{1 + ρ (t - a) π (a, t)} d a \end{matrix}$ and $\begin{matrix} (14) & F f (t) = \int_{t}^{\infty} \frac{m (a, t) f (a - t) e^{- \int_{a - t}^{a} μ (v, v + t - a) d v}}{1 + f (a - t) ϕ (a, t)} d a . \end{matrix}$

Let $L^{\infty} (0, \infty)$ denote the space of bounded functions on $(0, \infty)$ with the norm $\begin{matrix} ‖ u ‖_{\infty} = sup_{t > 0} | u (t) | . \end{matrix}$

According to (14), $F f (t) = 0$ for $t > A_{m}$ . Moreover, for $t ⩾ 0$ we have that $\begin{array}{l} (15) & K ρ (t) ⩽ Λ = sup_{t ⩾ 0} \int_{a_{m}}^{A_{m}} \frac{Q (a, t)}{π (a, t)} d a \end{array}$ and $\begin{array}{l} (16) & F f (t) ⩽ F_{0} = sup_{t ⩾ 0} \int_{t}^{A_{m}} \frac{m (a, t) e^{- \int_{a - t}^{a} μ (v, v + t - a) d v}}{ϕ (a, t)} d a, \end{array}$ where Λ and $F_{0}$ are finite numbers according to (8) and (9). In other words, $F f$ is a bounded function and the operator $K$ maps $L^{\infty} (0, \infty)$ into subspace of functions bounded by Λ.

Some other properties of the operator $K$ are given in the following lemma.
Lemma 2.2.
The operator $K$ is monotone and Lipschitz continuous on $L^{\infty} (0, \infty)$ .

Using monotonicity and Lipschitz continuity of $K$ , we come to the result that grants existence and uniqueness of solution to equation (12).
Theorem 2.3.
The equation ( 12 ) has a unique solution in $L^{\infty} (0, \infty)$ . Moreover, if the functions $Q (a, t)$ , $π (a, t)$ and $F f (t)$ are continuous with respect to t, then the solution to equation ( 10 ) is continuous.

Proofs of such assertions are quite standard and can be found in various references; see, e.g., [5,12,13]. Since our assumptions differ, we present proofs in the Appendix.
2.1. General upper and lower solutions

A nonnegative function $ρ^{+} \in L^{\infty} (0, \infty)$ is an upper solution to equation (12) if $\begin{array}{l} ρ^{+} (t) - K ρ^{+} (t) - F f (t) ⩾ 0 for t ⩾ 0 . \end{array}$ Similarly, a nonnegative function $ρ^{-} \in L^{\infty} (0, \infty)$ is a lower solution to equation (12) if $\begin{array}{l} ρ^{-} (t) - K ρ^{-} (t) - F f (t) ⩽ 0 for t ⩾ 0 . \end{array}$ The importance of this definition is that the upper and lower solutions give an upper and lower bounds for a solution ρ to equation (12). Namely, $\begin{array}{l} ρ^{-} (t) ⩽ ρ (t) ⩽ ρ^{+} (t) for t ⩾ 0 . \end{array}$ To prove this, suppose that $ρ^{-}$ is a lower solution to equation (12). We define an iterative sequence by $\begin{array}{l} (17) & ρ_{k + 1} = K ρ_{k} + F f and ρ_{0} = ρ^{-}, \end{array}$ where $K$ and $F f$ are defined by (13) and (14), respectively. By (17), we have that $\begin{array}{l} ρ_{1} = K ρ_{0} + F f ⩾ ρ_{0}, \end{array}$ and monotonicity of $K$ implies $\begin{matrix} ρ_{2} = K ρ_{1} + F f ⩾ K ρ_{0} + F f = ρ_{1} . \end{matrix}$ We continue the procedure and get $\begin{array}{l} ρ^{-} ⩽ ρ_{1} ⩽ ρ_{2} ⩽ \dots ⩽ ρ_{k} ⩽ \dots . \end{array}$ According to (15) and (16), $K ρ_{k}$ is bounded by Λ and $F f$ is a bounded by $F_{0}$ . Thus, the sequence ${ρ_{k}}$ is monotonically increasing and bounded from above and it converges to a fixed point ρ of the operator $K$ i.e. $ρ^{-} ⩽ ρ$ . In the similar way we can show that $ρ ⩽ ρ^{+}$ , where $ρ^{+}$ is an upper solution to equation (12).

This general result has useful applications. Namely, from the ecological perspective it is not always necessary to know the exact number of newborns. The practical problems sometimes indicate that it is more important to know the asymptotic behavior of the solution than the exact number of newborns at every point in time. The qualitative analysis of the solution can be done by using upper and lower bounds to solution ρ of equation (12), which are presented in the following theorem.

Theorem 2.4.
Let ρ be a solution to equation ( 12 ) and let $ρ^{}, ρ_{} \in L^{\infty} (0, \infty)$ satisfy $0 < c_{1} ⩽ ρ_{}, ρ^{} ⩽ c_{2} < \infty$ .
If $K ρ^{} (t) ⩽ ρ^{} (t)$ for $t > M^{} > 0$ , then there exists positive constants* $C_{1}$ and $α_{1}$ such that $\begin{array}{l} ρ (t) ⩽ ρ^{} (t) (1 + C_{1} e^{- α_{1} t}) for t ⩾ 0 . \end{array}$

If $K ρ_{} (t) ⩾ ρ_{} (t)$ for* $t > M_{} > 0$ , then there exists positive constants* $C_{2}$ and $α_{2}$ such that $\begin{array}{l} (18) & ρ (t) ⩾ ρ_{*} (t) (1 - C_{2} e^{- α_{2} t}) for t ⩾ 0 . \end{array}$

3. Time-independent case

Our goal is to study the asymptotic behavior of the number of newborns and of the total population in a changing environment, i.e., in the case when all functions that appear in the model are functions of age and time. However, solving the equation (12) or looking for its upper and lower solutions as required in Theorem 2.4 can be difficult. To make the problem simpler, we assume that the vital rates and the regulating function are time-independent. Thus, instead of the functions $Q (a, t)$ , $π (a, t)$ , $ϕ (a, t)$ and $ψ (a, t)$ defined by (4)–(7), in this section we use the functions $\begin{matrix} (19) & \begin{matrix} Q (a) = m (a) e^{- \int_{0}^{a} μ (v) d v}, \\ π (a) = \int_{0}^{a} \frac{μ (v) e^{- \int_{0}^{v} μ (s) d s}}{L (v)} d v, \\ ϕ_{1} (a, t) = \int_{a - t}^{a} \frac{μ (v) e^{- \int_{a - t}^{v} μ (s) d s}}{L (v)} d v, \\ ψ_{1} (a, t) = m (a) e^{- \int_{a - t}^{a} μ (v) d v} . \end{matrix} \end{matrix}$ Since $\begin{array}{l} \int_{0}^{a} μ (v) e^{- \int_{0}^{v} μ (s) d s} d v = - \int_{0}^{a} {(e^{- \int_{0}^{v} μ (s) d s})}^{'} d v = 1 - e^{- \int_{0}^{a} μ (s) d s}, \end{array}$ and $\begin{array}{l} e^{\int_{0}^{a - t} μ (v) d v} \int_{a - t}^{a} μ (v) e^{- \int_{0}^{v} μ (s) d s} d v = 1 - e^{- \int_{a - t}^{a} μ (s) d s}, \end{array}$ we can estimate $π (a)$ and $ϕ_{1} (a, t)$ as follows $\begin{array}{l} \frac{1}{L_{2}} (1 - e^{- \int_{0}^{a} μ (s) d s}) ⩽ π (a) ⩽ \frac{1}{L_{1}} (1 - e^{- \int_{0}^{a} μ (s) d s}) \end{array}$ and $\begin{array}{l} \frac{1}{L_{2}} (1 - e^{- \int_{a - t}^{a} μ (s) d s}) ⩽ ϕ_{1} (a, t) ⩽ \frac{1}{L_{1}} (1 - e^{- \int_{a - t}^{a} μ (s) d s}) . \end{array}$ In this way we obtain a simpler model compared to the one given by equations (1)–(3). The balance equation in the case of time-independent death rate and regulating function is $\begin{array}{l} (20) & \frac{\partial n (a, t)}{\partial t} + \frac{\partial n (a, t)}{\partial a} & = - μ (a) n (a, t) (1 + \frac{n (a, t)}{L (a)}), a, t > 0 . \end{array}$ The boundary condition for the time-independent birth rate is $\begin{array}{l} (21) & n (0, t) = \int_{0}^{\infty} m (a) n (a, t) d a, \end{array}$ and the initial condition is given by (3).

The next result is obtained directly from Theorem 2.1.

Corollary 3.1.
Let $n \in C^{1} ((0, \infty) \times (0, \infty))$ be a solution to the population problem ( 20 ), ( 21 ), ( 3 ). Then the function $ρ (t) = n (0, t)$ , $t ⩾ 0$ , satisfies the integral equation $\begin{array}{l} (22) & ρ (t) & = \int_{0}^{t} \frac{Q (a) ρ (t - a)}{1 + ρ (t - a) π (a)} d a + \int_{t}^{\infty} \frac{ψ_{1} (a, t) f (a - t)}{1 + f (a - t) ϕ_{1} (a, t)} d a . \end{array}$ For $a > 0$ , $n (a, t)$ can be found by $\begin{matrix} (23) & n (a, t) = \{\begin{matrix} \frac{ρ (t - a) e^{- \int_{0}^{a} μ (v) d v}}{1 + ρ (t - a) π (a)}, & a < t \\ \frac{f (a - t) e^{- \int_{a - t}^{a} μ (v) d v}}{1 + f (a - t) ϕ_{1} (a, t)}, & a ⩾ t . \end{matrix} \end{matrix}$

Writing equation (22) in the form $\begin{matrix} (24) & ρ (t) = K ρ (t) + F f (t), t ⩾ 0, \end{matrix}$ where $\begin{matrix} K ρ (t) = \int_{0}^{t} \frac{Q (a) ρ (t - a)}{1 + ρ (t - a) π (a)} d a \end{matrix}$ and $\begin{matrix} F f (t) = \int_{t}^{\infty} \frac{ψ_{1} (a, t) f (a - t)}{1 + f (a - t) ϕ_{1} (a, t)} d a, \end{matrix}$ leads us to the definition of the characteristic equation. Namely, in the linear age-structured time-independent model, the growth rate of population, denoted by r, is obtained by solving the Lotka-Euler characteristic equation $\begin{matrix} \int_{0}^{\infty} Q (a) e^{- r a} d a = 1 . \end{matrix}$ Behavior of the population is then estimated by the value of parameter r: for $r > 0$ population exponentially grows, for $r = 0$ it is constant and for $r < 0$ it exponentially declines.

Formally assuming that $ρ^{}$ is a constant solution to equation (24) for large t, we get the following characteristic equation $\begin{matrix} (25) & \int_{0}^{\infty} \frac{Q (a)}{1 + ρ^{} π (a)} d a = 1 . \end{matrix}$ We show that $ρ^{}$ is connected to the net reproductive rate $R_{0}$ , defined by $\begin{array}{l} R_{0} = \int_{0}^{\infty} Q (a) d a . \end{array}$ Moreover, we will prove that $ρ^{}$ plays a fundamental role in the description of the asymptotic behavior of functions $ρ (t)$ and $N (t)$ . The relation between $R_{0}$ and $ρ^{}$ is explained in the next lemma. Lemma 3.2.
Let Q be a continuous function. If* $R_{0} > 1$ , the integral equation ( 25 ) has a unique solution $ρ^{} > 0$ . If* $R_{0} ⩽ 1$ , equation ( 25 ) has no positive solution.

3.1. Asymptotics of the number of newborns and of the total population

In Theorem 2.4 we prove the existence of upper and lower bounds for the number of newborns in the general time-dependent case. Keeping the assumption that there is no temporal variability and using Lemma 3.2, we formulate the explicit upper and lower bounds for the functions $ρ (t)$ and $N (t)$ . In the following theorem we investigate the asymptotic behavior of the function $ρ (t)$ .

Theorem 3.3.
Let ρ be a solution to equation ( 24 ). If $R_{0} > 1$ , then there exist positive constants α and $C_{i}$ , $i = 1, 2$ , such that $\begin{array}{l} (26) & ρ^{} (1 - C_{1} e^{- α t}) ⩽ ρ (t) ⩽ ρ^{} (1 + C_{2} e^{- α t}) for large t, \end{array}$ where $ρ^{}$ is the solution to equation (* 25 ) and α does not depend on f.

If $R_{0} < 1$ , then there exist a positive constant C such that $\begin{array}{l} 0 ⩽ ρ (t) ⩽ C e^{- α t} for t ⩾ 0, \end{array}$ where α does not depend on f.

If $R_{0} = 1$ , then there exist a positive constant C such that $\begin{array}{l} 0 ⩽ ρ (t) ⩽ \frac{C}{1 + t} for t ⩾ 0 . \end{array}$

Using equation (23), the total population can be written as: $\begin{array}{l} (27) & N (t) & = \int_{0}^{t} \frac{ρ (t - a) e^{- \int_{0}^{a} μ (v) d v}}{1 + ρ (t - a) π (a)} d a + \int_{t}^{\infty} \frac{f (a - t) e^{- \int_{a - t}^{a} μ (v) d v}}{1 + f (a - t) ϕ_{1} (a, t)} d a . \end{array}$ Applying the previous result, we can study asymptotic representation of $N (t)$ . Corollary 3.4.
Let ρ be a solution to equation ( 24 ). If $R_{0} > 1$ , then $\begin{array}{l} N (t) = C ρ^{} + O (e^{- α t}) as t \to \infty, \end{array}$ where* $ρ^{}$ is the solution to equation (* 25 ), $C = \int_{0}^{A_{μ}} \frac{e^{- \int_{0}^{a} μ (v) d v}}{1 + ρ^{} π (a)} d a$ and α is a positive constant independent of f.*

If $R_{0} < 1$ , then $\begin{array}{l} N (t) = O (e^{- α t}) as t \to \infty, \end{array}$ where α is a positive constant independent of f.

If $R_{0} = 1$ , then $\begin{array}{l} N (t) = O (\frac{1}{1 + t}) as t \to \infty . \end{array}$

4. Estimates of the newborns and the total population in the time-dependent case

The large time behavior of a population in a temporally varying environment can be studied by examining asymptotic behavior of the solution ρ of the characteristic equation (12), as stated in Theorem 2.4. However, solving this equation can be difficult and impractical which prompts a different approach to the problem.

For a population in a constant environment, Theorem 3.3 and Corollary 3.4 give the exact upper and lower bounds for the functions ρ and N. In order to solve the original time-dependent problem, we formulate two auxiliary time-independent problems. One of them is the best case scenario (and it will provide an upper bound to a solution of the original problem) and the other is the worst case scenario (and it gives a lower bound). In what follows we will prove that by describing these corresponding problems we can find estimates of the solution to the original problem.

To formulate the auxiliary problems, let $a > 0$ and set $\begin{array}{l} m_{+} (a) = sup_{t ⩾ 0} m (a, t) and m_{-} (a) = inf_{t ⩾ 0} m (a, t), \\ μ_{+} (a) = sup_{t ⩾ 0} μ (a, t) and μ_{-} (a) = inf_{t ⩾ 0} μ (a, t), \\ L_{+} (a) = sup_{t ⩾ 0} L (a, t) and L_{-} (a) = inf_{t ⩾ 0} L (a, t) . \end{array}$ We define the functions $\begin{array}{l} Q_{\pm} (a) = m_{\pm} (a) e^{- \int_{0}^{a} μ_{\mp} (v) d v}, \\ π_{\pm} (a) = \frac{1}{L_{\mp} (a)} (1 - e^{- \int_{0}^{a} μ_{\pm} (v) d v}), \\ ψ_{\pm} (a, t) = m_{\pm} (a) e^{- \int_{a - t}^{a} μ_{\mp} (v) d v}, \\ ϕ_{\pm} (a, t) = \frac{1}{L_{\mp} (a)} (1 - e^{- \int_{a - t}^{a} μ_{\pm} (v) d v}), \end{array}$ and the integral operators $K_{\pm}$ and $F_{\pm}$ by $\begin{array}{l} K_{\pm} ρ (t) = \int_{0}^{t} \frac{Q_{\pm} (a) ρ (t - a)}{1 + ρ (t - a) π_{\mp} (a)} d a and F_{\pm} f (t) = \int_{t}^{\infty} \frac{ψ_{\pm} (a, t) f (a - t)}{1 + f (a - t) ϕ_{\mp} (a, t)} d a . \end{array}$ Since $Q_{-} ⩽ Q ⩽ Q_{+}$ and $π_{-} ⩽ π ⩽ π_{+}$ , monotonicity of the operator $K$ implies $\begin{array}{l} K_{-} ρ (t) ⩽ K ρ (t) ⩽ K_{+} ρ (t) . \end{array}$ Due to the fact that $ϕ_{-} ⩽ ϕ ⩽ ϕ_{+}$ and $ψ_{-} ⩽ ψ ⩽ ψ_{+}$ , we have that $\begin{array}{l} F_{-} f (t) ⩽ F f (t) ⩽ F_{+} f (t) . \end{array}$ This leads us to the auxiliary problems: $\begin{array}{l} ρ_{-} = K_{-} ρ_{-} + F_{-} f and ρ_{+} = K_{+} ρ_{+} + F_{+} f . \end{array}$ By definition of the operators $F_{\pm} f$ , it follows that $ρ_{-} ⩽ K ρ_{-} + F f$ and $ρ_{+} ⩾ K ρ_{+} + F f$ . Hence, $ρ_{-}$ is a lower solution and $ρ_{+}$ is an upper solution to equation (12).

The two-side estimates of solution ρ to equation (12) are presented in the following theorem.

Theorem 4.1.
Let ρ be a solution to equation ( 12 ). If $\int_{0}^{\infty} Q_{-} (a) d a > 1$ , then there exist positive constants $α_{\pm}$ and $C_{\pm}$ , such that $\begin{array}{l} ρ_{-}^{} (1 - C_{-} e^{- α_{-} t}) ⩽ ρ (t) ⩽ ρ_{+}^{} (1 + C_{+} e^{- α_{+} t}) for large t \end{array}$ where $ρ_{\pm}^{}$ are solutions to the equations* $\begin{array}{l} (28) & \int_{0}^{\infty} \frac{Q_{\pm} (a)}{1 + ρ_{\pm}^{} π_{\mp} (a)} d a = 1 \end{array}$ and the constants* $α_{\pm}$ are independent of f.

If $\int_{0}^{\infty} Q_{+} (a) d a < 1$ , then there exist positive constants α and C, such that $\begin{array}{l} 0 ⩽ ρ (t) ⩽ C e^{- α t} for large t \end{array}$ and the constant α is independent of f.

If $\int_{0}^{\infty} Q_{+} (a) d a = 1$ , then there exists positive constant C, such that $\begin{array}{l} 0 ⩽ ρ (t) ⩽ \frac{C}{1 + t} for large t . \end{array}$

For the total population $N (t)$ a similar result holds. Theorem 4.2.
If $\int_{0}^{\infty} Q_{-} (a) d a > 1$ , then there exist positive constants $α_{\pm}$ independent of f, such that $\begin{array}{l} (29) & C_{-} ρ_{-}^{} + O (e^{- α_{-} t}) ⩽ N (t) ⩽ C_{+} ρ_{+}^{} + O (e^{- α_{+} t}) for large t, \end{array}$ where $ρ_{\pm}^{}$ are solutions to the equations (* 28 ) and the constants $C_{\pm}$ are defined by $\begin{matrix} C_{-} = \int_{0}^{A_{μ}} \frac{e^{- \int_{0}^{a} μ_{+} (v) d v}}{1 + ρ_{-}^{} π_{+} (a)} d a and C_{+} = \int_{0}^{A_{μ}} \frac{e^{- \int_{0}^{a} μ_{-} (v) d v}}{1 + ρ_{+}^{} π_{-} (a)} d a . \end{matrix}$

If $\int_{0}^{\infty} Q_{+} (a) d a < 1$ , then there exist a positive constant α independent of f, such that $\begin{array}{l} (30) & N (t) ⩽ O (e^{- α t}) for large t . \end{array}$

If $\int_{0}^{\infty} Q_{+} (a) d a = 1$ , then the total population $N (t)$ satisfies $\begin{array}{l} (31) & N (t) ⩽ O (\frac{1}{1 + t}) for large t . \end{array}$

5. Asymptotics of solution in the case of periodic vital rates

In most cases, natural environments are positively autocorrelated. This means that the temporal changes follow some pattern and can be presented using periodic functions. The change of seasons or daily oscillations in temperature, light and humidity are some examples of this type of variability. We include it in the model and assume that the variation has effect only on the birth rate, leaving the death rate time-independent. For simplicity, the regulating function is a constant. In other words, the vital rates and the regulating function are given by $\begin{array}{l} (32) & m (a, t) = m_{0} (a) + ε cos (A t) m_{1} (a), a, t > 0, \\ (33) & μ (a, t) = μ (a), \\ (34) & L (a, t) \equiv L, L > 0, \end{array}$ where $m_{0}$ and $m_{1}$ satisfy assumption (i) and $A, ε > 0$ . For brevity, we use the notation $\begin{array}{l} Q_{i} (a) = m_{i} (a) e^{- \int_{0}^{a} μ (v) d v}, i = 1, 2 and π (a) = \frac{1}{L} (1 - e^{- \int_{0}^{a} μ (v) d v}) . \end{array}$ Under these assumptions, we obtain an explicit asymptotic representation of the number of newborns.

Theorem 5.1.
Let ρ be a solution to equation ( 12 ) where the vital rates and the regulating function are given by ( 32 ), ( 33 ) and ( 34 ). If $\int_{0}^{\infty} Q_{0} (a) d a > 1$ , then $\begin{array}{l} ρ (t) & = ρ_{0}^{} + ε (c_{1} cos A t + d_{1} sin A t) + ε^{2} (k_{2} + c_{2} cos 2 A t + d_{2} sin 2 A t) + O (ε^{3}), \end{array}$ where* $ρ_{0}^{}$ is a positive solution to the equation* $\begin{matrix} (35) & \int_{0}^{\infty} \frac{Q_{0} (a)}{1 + ρ_{0}^{} π (a)} d a = 1, \end{matrix}$ and the parameters* $c_{1}$ , $d_{1}$ , $c_{2}$ , $d_{2}$ , and $k_{2}$ depend on m, μ, L and A.

Using Theorem 5.1 we obtain an estimation of the average number of newborns $ρ_{av}$ and the average total population $N_{av}$ , where $ρ_{av}$ is defined by $\begin{array}{l} ρ_{av} = \frac{A}{2 π} \int_{0}^{2 π / A} ρ (t) d t . \end{array}$ Corollary 5.2.
Let the assumptions of Theorem 5.1 hold. If $\int_{a_{m}}^{A_{m}} Q_{0} (a) d a > 1$ and $ε > 0$ is a small number, then the average number of newborns is $\begin{array}{l} (36) & ρ_{av} = ρ_{0}^{} + ε^{2} k_{2} + O (ε^{3}) . \end{array}$ Moreover, the average total population is* $\begin{array}{l} (37) & N_{av} = (ρ_{0}^{*} + ε^{2} k_{2}) \int_{0}^{A_{μ}} \frac{e^{- \int_{0}^{a} μ (v) d v}}{1 + ρ_{av} π (a)} d a + O (ε^{3}) . \end{array}$

6. Improvement of population stability due to a variable environment

It is already noted in [17,24,29] that changes in the environment can be beneficial or detrimental for population growth depending on their pattern and on the population responsiveness. According to Theorem 5.1 and Corollary 5.2, the number of newborns and the total population depend on the vital rates and on the frequency of oscillation. In the following example, we use four different life-histories and study how change of the frequency of oscillation affects population growth.

Table 1
Life histories of ursus, calidris, ectotherm and insect

Age class Ursus Calidris Ectotherm Insect

s m s m s m s m

1 0,67 0 0,32 0 0,13 0 0,54 0

2 0,75 0 0,78 1 0,2 1 0,52 0

3 0,82 0 0,72 1 0,17 30 0,49 0

4 0,9 0,5 0,66 1 0,14 30 0,47 0

5 0,86 0,5 0,6 1 0,11 30 0,45 0

6 0,82 0,5 0,54 1 0,09 30 0,42 0

7 0,78 0,5 0,48 1 0,06 30 0,4 0

8 0,74 0,5 0,42 1 0,03 30 0,38 83,33

9 0,7 0,5 0,36 1 0,01 30 0,36 166,67

10 0,65 0,5 0,3 1 0 30 0,34 250

11 0,61 0,5 0,24 1 0,32 333,33

12 0,57 0,5 0,18 1 0,3 416,67

13 0,53 0,5 0,12 1 0,27 500

14 0,49 0,5 0,06 1 0,25 433,33

15 0,45 0,5 0,01 1 0,23 400

16 0,41 0,5 0 1 0,21 366,67

17 0,37 0,5 0,19 333,33

18 0,33 0,5 0,17 300

19 0,29 0,5 0,15 266,67

20 0,25 0,5 0,13 233,33

21 0,21 0,5 0,11 200

22 0,16 0,5 0,09 166,67

23 0,12 0,5 0,06 133,33

24 0,08 0,5 0,04 100

25 0,04 0,5 0,02 66,67

26 0,01 0,5 0,01 33,33

27 0 0,5 0 0

Age class	Ursus	Calidris	Ectotherm	Insect
1	0,67	0	0,32	0	0,13	0	0,54	0
2	0,75	0	0,78	1	0,2	1	0,52	0
3	0,82	0	0,72	1	0,17	30	0,49	0
4	0,9	0,5	0,66	1	0,14	30	0,47	0
5	0,86	0,5	0,6	1	0,11	30	0,45	0
6	0,82	0,5	0,54	1	0,09	30	0,42	0
7	0,78	0,5	0,48	1	0,06	30	0,4	0
8	0,74	0,5	0,42	1	0,03	30	0,38	83,33
9	0,7	0,5	0,36	1	0,01	30	0,36	166,67
10	0,65	0,5	0,3	1	0	30	0,34	250
11	0,61	0,5	0,24	1			0,32	333,33
12	0,57	0,5	0,18	1			0,3	416,67
13	0,53	0,5	0,12	1			0,27	500
14	0,49	0,5	0,06	1			0,25	433,33
15	0,45	0,5	0,01	1			0,23	400
16	0,41	0,5	0	1			0,21	366,67
17	0,37	0,5					0,19	333,33
18	0,33	0,5					0,17	300
19	0,29	0,5					0,15	266,67
20	0,25	0,5					0,13	233,33
21	0,21	0,5					0,11	200
22	0,16	0,5					0,09	166,67
23	0,12	0,5					0,06	133,33
24	0,08	0,5					0,04	100
25	0,04	0,5					0,02	66,67
26	0,01	0,5					0,01	33,33
27	0	0,5					0	0

Table 1 contains the data for the birth rate $m_{0} (a)$ and survival rate $s (a) = e^{- \int_{a - 1}^{a} μ (v) d v}$ for ursus, calidris, ectotherm and insect. These synthetic life-histories are constructed to cover a wide spectra of species and they are used to indicate to which family or order a species with such a life-history belongs; see [15]. Notice that this is not a real-life data, but a construct that is used for a study of relations depending on the life-history.

For the purpose of this example, using (32), we define the time-dependent birth rate by setting $m_{1} (a) = e^{- 0, 1 a}$ . We assume that the functions $m_{0} (a)$ and $m_{1} (a)$ have the same support. The regulating function has constant value $L = 100$ and $ε = 0, 01$ .

The generation time (i.e., the average age of having first offspring) is calculated according to formula $\begin{matrix} T = \int_{0}^{\infty} e^{- r a} m_{0} (a) s (a) a d a, \end{matrix}$ where r is the growth rate of population in the constant environment, i.e., a solution to the Lotka-Euler characteristic equation $\begin{matrix} \int_{0}^{\infty} e^{- r a} m_{0} (a) s (a) d a = 1 . \end{matrix}$

Based on the data from the Table 1, the net reproductive rate $R_{0}$ is less than 1 in all four cases. To apply Theorem 5.1, $R_{0} > 1$ must be fulfilled. Therefore, we scale $m_{0} (a)$ . For ursus we multiply by 1.5, for calidris by 2, for ectotherm and for insect by 10. Lemma 3.2 and Corollary 5.2 provide values for $ρ_{0}^{*}$ , the average number of newborns $ρ_{av}$ and the average total population $N_{av}$ . The values are given in the Table 2.

Table 2

The net reproductive rate, generation time, number of newborn and average total population

	Ursus	Calidris	Ectotherm	Insect
$R_{0}$	1,17	1,37	1,78	1,76
T	6,03	2,68	2,44	8,93
$ρ_{0}^{*}$	23,25	45,16	79,45	76,55
$N_{av}$	64,36	33,63	7,49	55,81

In all cases except for ursus, the results from the Table 2 show that the total population is smaller than the number of newborns. A possible explanation for this is in the low survival rate for calidris, ectotherm and insects and in the fact that the Table 1 represents vital rates in discrete time. As we see it, the survival rate of newborns is only 0,13 for ectotherm compared to 0,67 for ursus. This trend is present in other age classes which means that very few newborn ectotherms survive long enough to reach the next age-class and even fewer of them survive to reach the fertile age.

According to (36), the average number of newborns in a variable environment will be changed for the amount equal to $ε^{2} k_{2}$ . Using the data from the Table 1 and (57), we plot $k_{2}$ for ursus, calidris, ectotherm and insect. Results are presented in the Fig. 1. Several conclusions follow.

Fig. 1.

$k_{2}$ as a function of frequency A of oscillation.

First, $k_{2} (A) > 0$ only for small A implying that only low frequency oscillations may be beneficial for population growth for any observed life-histories. All other oscillations are detrimental for population growth due to the fact that $k_{2} (A) < 0$ for larger A.

Second, since $ε > 0$ is a small number, (36) asserts that $ρ_{0}^{*} + ε^{2} k_{2} (A) > 0$ for sufficiently large $ρ_{0}^{*}$ and all $k_{2} (A)$ . This can be related to [24], where the authors argue that temporal variability can cause extinction if a population is very small or if the amplitude of oscillation is large. In other words, if the average number of newborns is above certain threshold value, oscillations can decrease population growth, but they cannot be responsible for extinction. How changes in $ε > 0$ affect the threshold were studied in [9].

Third, according to (57), $k_{2}$ depends on the vital rates. Comparing the values of $k_{2}$ from Fig. 1 for different life-histories, we see that ursus and calidris are more sensitive to the environmental changes than ectotherm and insect. A deeper analysis of relation between the life-histories and frequencies of the environmental change would be useful.

7. Discussion

There are many different population models, but only a small number of them study the interplay between the age-structure, density-dependence and temporal variation. In most cases, the structured logistic population models presuppose that the vital rates are time-independent and the logistic term is a function of the total population (see, e.g., [4,32]). Following considerations about the carrying capacity and the ecological implications of its definition (see, e.g., [22]), we include time and age-dependent logistic term instead of the constant carrying capacity. The specific form of the logistic term in equation (1) reflects that competition occurs only between members of the same age class and that environmental factors change with age and time. The limitations imposed by the environment are introduced through the regulating function $L (a, t)$ . The regulating function itself does not represent a limit of population density. This is in contrast to the Verhulst model where the logistic term represents intra-specific competition (i.e. competition on the population level) and the constant carrying capacity is a nonzero equilibrium point.

Biologically, our choice of the logistic term is partially justified by the facts that, depending on the species and on the resource, individuals from different age-classes might not compete. Therefore, an individual finds its competitors within its own age class. In addition to this, we assumed that competition has influence only on death rate leaving the birth rate unaffected.

Although it is very restricted, the logistic term $\frac{μ (a, t) n^{2} (a, t)}{L (a, t)}$ can be used as a starting point in the study of the combined influences of the age-structure, density-dependence and temporal variability. Namely, a solution to the linear age-structured model (obtained from (1) by formally assuming that the logistic term is identically equal to zero) is the exponential function, while solutions to the various density-dependent models, including model (1)–(3), are bounded functions. As a future research, the solution to (1)–(3) can be used to formulate estimates of a solution to a more general model.

We use the method of upper and lower solutions known in the theory of integral equations and prove that a solution $ρ (t)$ to the equation (12) can be bounded from above and below by certain bounded function. Population growth can be studied and possible extinctions or explosions can be predicted by examining mentioned boundaries.

In the case when the vital rates and the regulating function are time-independent, we prove that solution $ρ^{*}$ to the characteristic equation (25) plays a fundamental role in estimating the number of newborns and the total population. Namely, for large t the following holds: $\begin{array}{l} ρ (t) = \{\begin{matrix} ρ^{*} + O (e^{- α t}) & if R_{0} > 1, \\ O (\frac{1}{1 + t}) & if R_{0} = 1, \\ O (e^{- α t}) & if R_{0} < 1, \end{matrix} \end{array}$ where α is a positive constant independent of f. Similarly, the total population satisfies $\begin{array}{l} N (t) = \{\begin{matrix} C ρ^{*} + O (e^{- α t}) & if R_{0} > 1, \\ O (\frac{1}{1 + t}) & if R_{0} = 1, \\ O (e^{- α t}) & if R_{0} < 1, \end{matrix} \end{array}$ where C and α are positive constants and α is independent of f.

We use these results to study general time-dependent model (1)–(3). Namely, taking the supremum and infimum of the model parameters, we define their time-independent counterparts in the best and worst case. Solutions to these time-independent problems are possible to obtain and analyze, as we have seen. The best case scenario defines upper bounds to the original problem, while the worst case scenario gives lower bounds. Thus, if $ρ (t)$ is the solution to the original time-dependent problem and $ρ_{-} (t)$ and $ρ_{+} (t)$ are the worst and the best case solutions, it follows that $\begin{matrix} ρ_{-} (t) ⩽ ρ (t) ⩽ ρ_{+} (t) . \end{matrix}$

The conclusion is that regardless of the pattern of the environmental change, the number of newborns and the total population lies within the boundaries defined in the best and worst case scenarios and the extinction risk can be predicted in accordance to the net reproductive rates in the two extreme situations. In the case when $R_{0}^{-} < 1 < R_{0}^{+}$ , analysis in Section 4 does not provide any conclusion. The analysis can be improved by taking periodic problems to formulate upper and lower boundaries to the solutions; see for example [16]. In the epidemiological models such as [1] and [31], a similar problem is resolved by using the threshold value that determines permanence of a solution.

Majority of natural environments are predictable in the sense that temporal changes often exhibit recurring behavior. Hence, it is reasonable to assume that the vital rates and the environmental factors are changing periodically. For simplicity, suppose that environmental variability has influence only on the birth rate, leaving the death rate and the regulating function unaffected. If we additionally suppose that the average net reproductive rate is strictly larger than one, then the average number of newborns can be found by $\begin{array}{l} ρ_{av} = ρ_{0}^{*} + ε^{2} k_{2} + O (ε^{3}) \end{array}$ and the average total population is $\begin{matrix} N_{av} = (ρ_{0}^{*} + ε^{2} k_{2}) \int_{0}^{A_{μ}} \frac{e^{- \int_{0}^{a} μ (v) d v}}{1 + ρ_{av} π (a)} d a + O (ε^{3}) . \end{matrix}$ This implies that oscillations in the birth rate, caused by changes in the environment, have effects on the number of newborns and on the total population and on their average values.

A more detailed analysis in Section 6 revealed that $k_{2}$ can change sign for different frequencies of oscillation, implying that some oscillations are beneficial for population growth, and the others are detrimental. Conclusions for the logistic model partially differ from the results that hold for the linear discrete-time model and the linear continuous-time model, [29] and [17], respectively. In the logistic model, low-frequency changes of the environment may be beneficial for population growth regardless of species, which is in sharp contrast to the mentioned linear models. A higher frequency environmental variability seems to have negative effect on population growth for all observed species. The intensity of the effect depends on the species and it is stronger for ursus and calidris than for ectotherm and insect. Thus, we are inclined to think that the life-history plays important role and must be considered together with the age-structure and the environmental variation in the study of population dynamics.

Boyce and Daley [2], May [20] and Roughgarden [25] claim that the environmental variation keeps population numbers below its carrying capacity. Our conclusion is that this is true for the frequencies that give negative $k_{2}$ . Roughgarden [26] argues that fluctuations in the environment cause discrepancy between the population size and the carrying capacity and that a relation between population’s ability to track fluctuating resources and predictability of the environment can be found. We come to the similar result if we interpret $k_{2}$ as a measure of population’s response to the environmental changes. However, in the age-structured population model, the relation is more complex and it depends on the demography, life-history and the environmental variation, while in the unstructured models of May and Roughgarden it depends only on the intrinsic growth rate and the frequency of oscillation.

As a final remark, we would like to point out that model (1)–(3) is based on the assumption that competition occurs only within age-class. This is partially a technical assumption, although we found biological explanation for it. Besides, in the continuous age-structured models there is no discontinuities between age-classes like those present in the discrete age-structured models. This reflects on the vital rates and on the competition strength between individuals of nearly same age. In other words, due to continuity, the difference in the model parameters for individuals of nearly same age is very small. In the more realistic settings, one could use for example $\begin{matrix} μ (a, t, ν (t)), where ν (t) = \int_{0}^{\infty} α (a, t) n (a, t) d a, \end{matrix}$ and $α (a, t)$ is a weight function, instead of the function $- \frac{μ (a, t) n^{2} (a, t)}{L (a, t)}$ , to include competition between individuals of different age.

Footnotes

For completing the proof of Theorem 2.4, we need the following lemma and its corollary.

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