- Research
- Open Access
Estimating effective boundaries of population growth in a variable environment
- Vladimir Kozlov^{1},
- Sonja Radosavljevic^{1}Email authorView ORCID ID profile,
- Bengt Ove Turesson^{1} and
- Uno Wennergren^{2}
- Received: 10 March 2016
- Accepted: 14 September 2016
- Published: 22 September 2016
Abstract
We study the impact of age-structure and temporal environmental variability on the persistence of populations. We use a linear age-structured model with time-dependent vital rates. It is the same as the one presented by Chipot in (Arch. Ration. Mech. Anal. 82(1):13-25, 1983), but the assumptions on the vital rates are slightly different. Our main interest is in describing the large-time behavior of a population provided that we know its initial distribution and transient vital rates. Using upper and lower solutions for the characteristic equation, we define time-dependent upper and lower boundaries for a solution in a constant environment. Moreover, we estimate solutions for the general time-dependent case and also for a special case when the environment is changing periodically.
Keywords
- age-structure
- time-dependency
- environmental variability
- population growth
- upper and lower bounds
- periodic oscillations
MSC
- 35B40
- 35C15
- 92D25
1 Introduction
We live in the age when the number of rare or nearly extinct species is growing daily. Climate change and environmental pollution caused by human activity have profound impact on the extinction risk for many species; see, e.g., [2]. In order to avert the possible extinction and preserve the diversity in nature, we need to understand how temporal environmental changes influence the birth and death rates of individuals in a population, and by influencing the birth and death rates, how these changes influence population growth.
When it comes to the permanence of a population, safety is not always in numbers because environmental variation affects a population regardless of its size. Moreover, one of the main concerns for conservation biologists is in endangered species. This means that small populations are encountered in most cases, and for them demography plays a much more prominent role in comparison to large populations. In other words, the risk of extinction due to demography for small populations is high.
The conclusion that arises is that both demography and temporal environmental variability need to be considered when we set up a population model. Demography is traditionally introduced in population models through age-structure. One of the first such models was developed by Sharpe and Lotka [3] in 1911, McKendrick [4] in 1926, and von Foerster [5] in 1959. This model describes population dynamics as a linear process. A central place has the age-class density function \(n(a,t)\), where a is the age of an individual and t is the time. A population grows (or declines) exponentially, which is defined by age-dependent birth and death rates through the Lotka-Euler characteristic equation. A solution to the characteristic equation is related to the net reproductive rate and it uniquely determines population dynamics. For more details and analysis of the continuous linear model we refer the reader to [6–9], and [10]. There is also interest for studying nonlinear age-dependent models, in which case the vital rates depend on the age-class and on the population density; see for instance [9, 11] and [12]. An analysis of discrete population models can be found in [13].
From the ecological point of view, it is crucially important to study the interplay between age-structure and environmental variation and describe how combination of these two factors influence population growth. Besides, based on the information as regards transient vital rates, a model should be able to predict population dynamics for any time. This is translated into an analysis of the large-time behavior of a solution to an age-structured time-dependent model.
The linear time-independent model, mentioned above, is easy to use and on many occasions it gives satisfactory results. Unfortunately, the model lacks the ability to deal with environmental fluctuations and its effects on population growth. To encompass environmental variability, time-dependency must be included to a model. We refer to Chipot in [1] where the linear theory developed by Sharpe and Lotka and McKendrick is extended through time-dependent vital rates. Under certain assumptions on the vital rates, using a fixed point argument, he proved that the model has a unique nonnegative solution. Cushing in [14] studied existence of time-periodic solutions to the model under specific assumptions on the vital rates.
In order to investigate how transient time-dependent vital rates affect dynamics of an age-structured population, we use the same model as Chipot in [1]. Under slightly different conditions for the vital rates we prove the existence and uniqueness of a solution. However, the main part of our work is dedicated to the analysis of its large-time behavior.
Since the vital rates are now time dependent, the characteristic equation is difficult, if not impossible, to solve analytically. Ecologically, not knowing the exact solution of the problem can be compensated for by considering the boundaries within which a solution fluctuates in time. Very low boundaries indicate the risk of extinction and very high boundaries can imply unrestrained increase.
Using the method of upper and lower solutions for integral equations makes solving the characteristic equation redundant. We obtain upper and lower solutions for the characteristic equation instead and use them to determine upper and lower bounds for the number of newborns and the total population. These upper and lower bounds are time-dependent functions that correspond to the best case scenario and to the worst case scenario, respectively. Provided that the initial distribution of a population and the vital rates are known, we estimate the population density in the worst and in the best case scenarios. Naturally, in all other cases the population density lies between boundaries given by the population densities in these two extreme cases. As in the time-independent case, the population exhibits exponential growth or decline, depending on the vital rates.
Temporal environmental variability quite often presuppose periodic changes. Under the assumption that the birth rate is a time-periodic function and the death rate is a time-independent function, we obtain an exact upper and lower bound for the number of newborns and the total population. Tuljapurkar in [15] presents a similar study for a discrete linear population model and proves that population growth is governed by the average vital rates. He claims that the growth rate is increased by oscillations with periods near the generation time and decreased by oscillations with much shorter or much longer periods.
Our analysis also concludes that the average vital rates determine population growth. However, if a population has zero intrinsic growth rate, the frequency of oscillation can cause growth or decline, depending on the life history. Oscillations with very low frequencies are detrimental for population growth for all observed life histories. Unlike Tuljapurkar, we show that if the period is comparable to the generation time, one needs to consider life history as well since different species having different responses to changes in the environment. This observation implies that there is a deeper connection between age-structure and time dependency that should be investigated.
2 Age-structured model in a variable environment
3 General upper and lower bounds
In analogy to the analysis that followed from the Lotka-Euler characteristic equation, we base our predictions of population growth on equation (13) and the function σ. Since solving equation (13) can be a difficult task, we agree to the following trade-off: instead of solving the original problem (i.e., finding a fixed point of equation (6)), we are looking for upper and lower bounds of equation (6).
Definition 3.1
The next result is about upper solutions to equation (6).
Theorem 3.2
Proof
The following theorem deals with the problem of finding a lower solution to equation (6). In combination with the previous result, it allows us to describe boundaries for the density of newborns for large time t.
Theorem 3.3
Proof
Theorems 3.2 and 3.3 give us the following upper and lower bounds for the number of newborns.
Corollary 3.4
Theorem 3.5
Proof
4 Existence of the function σ
In order to prove the main theorem of this section, we need the following lemma.
Lemma 4.1
Proof
The solution h is a limit of the sequence \(h_{n+1}=Hh_{n}\), \(h_{0}=0\) as \(n\rightarrow\infty\). Its positivity follows from the fact that the sequence is monotonically increasing and each term is nonnegative on \([M,\infty)\).
Finally, for \(t=M\), \(Hh(M)\) is equal to the left-hand side of (23), which completes the proof. □
Theorem 4.2
Suppose that Q is differentiable with respect to t and \(Q'_{t}\) is bounded on \(\mathbb{R}^{+}\times \mathbb{R}^{+}\). If \(\gamma\in L^{\infty}(0,M)\), where \(M\geq A_{m}\), satisfies (23), then the characteristic equation (13) has a unique solution \(\sigma\in L^{\infty}(0,\infty)\) such that \(\sigma =\gamma\) on \([0,M]\).
Proof
5 Upper and lower bounds through time-independent models
Theorem 5.1
Using Theorem 5.1, we obtain the estimates for the total population.
Corollary 5.2
Naturally, estimates of the number of newborns and the total population obtained by Theorem 3.2, Theorem 3.3, and Theorem 3.5 are finer than the estimates provided by Theorem 5.1 and Corollary 5.2, but they are harder to get. For practical purposes it is often enough to have a prognosis for population growth in the best and worst case, which makes the upper and lower bounds defined in Theorem 5.1 and Corollary 5.2 a useful tool for predicting the fate of a population.
6 Periodical changes of the environment
Quite often populations live in periodically changing habitats. Our general model allows any kind of temporal environmental change, but for practical reasons we study periodical changes in detail. We assume the birth rate is a periodic function with respect to time and the death rate is a time-independent function. Under these stronger conditions, we can find explicit forms of the upper and lower bounds for the number of newborns and for the total population.
Theorem 6.1
Proof
It is clear now that (34) is negative only if (38) is negative. Since \(\int_{t-a}^{t}\sigma_{1}(\tau)\,d\tau\) and \(\int _{t-a}^{t}\sigma_{2}(\tau)\,d\tau\) are bounded functions for all t, it follows that the term (34) is negative provided that C is sufficiently large.
In a similar way we prove the left-hand side inequality of (33). □
Corollary 6.2
The behavior of population growth in cyclic environments for discrete time has already been analyzed. As expected, our results correspond to the main results presented by Tuljapurkar in [15]. According to Corollary 6.2, and due to the fact that \(\varepsilon >0\) is a small number, the large-time behavior of \(N(t)\) is determined by the parameter \(k_{0}\). For negative values of \(k_{0}\), the total population is decreasing and extinction of the population is imminent. On the other hand, a positive \(k_{0}\) yields population growth and survival of population is granted. In the borderline case when \(k_{0}=0\), the behavior of the total population is determined by the parameter \(k_{2}\). We observe population growth for positive \(k_{2}\) and population decline for negative \(k_{2}\).
- (1)
If \(R_{0}<1\), then \(k_{0}<0\) and \(N(t)\rightarrow0\) as \(t\rightarrow \infty\) for small ε.
- (2)
If \(R_{0}>1\), then \(k_{0}>0\) and \(N(t)\rightarrow\infty\) as \(t\rightarrow\infty\) for small ε.
- (3)
If \(R_{0}=1\), then \(k_{0}=0\) and \(N(t)\rightarrow\infty\) or \(N(t)\rightarrow0\) as \(t\rightarrow\infty\), depending on the sign of \(k_{2}\).
7 Improvement of stability due to a variable environment
According to equation (32), for fixed birth and death rates, the parameter \(k_{2}\) can change its sign for different values of A. By Theorem 6.1 and Corollary 6.2 it is obvious that changes in \(k_{2}\) reflect on population growth either by promoting it (for \(k_{2}>0\)) or by dampening it (for \(k_{2}<0\)). In order to get some insight into the behavior of the parameter \(k_{2}\) and its effect on the persistence of the population, we use the real life data for the vital rates for four different species. Although \(k_{2}\) depends not only on the frequency of oscillation, but also on the vital rates, here we will focus on changes in the frequency.
Life histories
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 |
The vital rates in Table 1 are the birth rate m and the survival probability s, given by \(s(a)=e^{-\int_{a-1}^{a}\mu(v)\,dv}\) for \(a\in[1, A_{\mu}]\). We define the periodic birth rate using (30). For simplicity, we assume that \(\gamma=0\).
We use equation (31) to compute \(k_{0}\) for all species and (32) to plot graphs of the function \(k_{2}(A)\) for the frequencies \(A\in[0.01,10]\) for all four life histories.
Characteristic parameters
Ursus | Calidris | Ectotherm | Insect | |
---|---|---|---|---|
\(k_{0}\) | −0.0409 | −0.1234 | −0.6569 | −0.1214 |
\(R_{0}\) | 0.78 | 0.6874 | 0.1789 | 0.3529 |
\(A_{T}\) | 1.36 | 1.95 | 2.28 | 0.59 |
T | 4.59 | 3.21 | 2.75 | 10.54 |
We would like to point out that our conclusions are valid for the vital rates given in Table 1. From equation (32) it follows that \(k_{2}\) is a function of the generation time T. Since the generation time T depends on life history, it is worth investigating how changes in life history will reflect on the population growth in a periodic environment.
8 Discussion
We studied a population model derived from the linear age-structured time-dependent model presented by Chipot in [1]. The difference between these two models is in the properties of the vital rates. From the biological point of view, this approach is justified since we need a tool for analyzing population growth and predicting extinctions or explosions of population due to temporally changing environment. We obtained several new results which are in accordance with the classical theory developed for the time-independent or for the discrete models.
Contrary to the Lotka-Euler characteristic equation (12) in the classical time-independent model, the characteristic equation in the time-dependent model is given by (13), which emphasizes the fact that we study the interplay between age- and time-dependent vital rates and population growth.
Since ε is a small number, the dominant role in the asymptotic behavior of the total population \(N(t)\) belongs to the parameter \(k_{0}\). For \(k_{0}>0\), the population is growing, while for \(k_{0}<0\) it is declining. In the special case when \(k_{0}=0\), the large-time behavior of the population is entirely determined by the sign of \(k_{2}(A)\), which is, according to (32), a function depending on the frequency A.
Using real life data we came to the following conclusions. If the period of oscillation is much longer than the generation time, the oscillation decreases the growth of a population in all observed cases. On the other hand, with oscillations whose periods are comparable to the generation time, we must be more careful. If the period is shorter than the generation time, changes in the environment increase the population growth for ursus and calidris and reduce it for ectotherm and insect. However, if the period is longer that the generation time, populations of ursus and calidris will decrease and the populations of ectotherm and insect will increase.
Our results expand those from a discrete model in Tuljapurkar [15]. The differences are in the fact that he deals with the discrete population model and does not use different life histories. Moreover, he claims that oscillations with period either much longer or shorter that the generation time are detrimental for population growth, and oscillations with period comparable to the generation time are beneficial for population growth. We show that for periods comparable to the generation time, there does not exist an unanimous effect on population growth for all species and all period lengths.
Finally, according to (32), \(k_{2}\) is a function of the vital rates and frequency. It is worth investigating how changes in life history reflect on the behavior of the function \(k_{2}\).
Declarations
Acknowledgements
The authors are thankful to the anonymous referees for useful comments and for pointing out interesting literature that tackles similar topics; see [17, 18] and [19].
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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