## Population Dynamics and Growth Models

Population growth rate is defined as the change in number of individuals in the population through a certain amount of time. Changes in abundance often are assessed at the population level, because that is where the effects of biological and environmental conditions are most evident. Later the modeling of metapopulations (groups of populations that share individuals) will be discussed. Often ecologists measure population changes using density (the number of individuals per area) because interactions among individuals and between an individual and its environment are more affected by density than actual Uniform Random Aggregated

Figure 1 Various forms of dispersion in space (uniform, random, and aggregated) are depicted using red stars as individual organisms. In reality, there is a continuum of dispersion from the extremely uniform to greatly aggregated, with random in between these two forms.

population abundance. Changes in abundance of a population, called population dynamics, are caused by many factors, but at the most fundamental level the number of individuals in the population at some later time period (Nt +1) is simply the number of individuals at the beginning of the time period (N) plus the number of births (B) and immigrants (I) minus the number of deaths (D) and emigrants (E) that occur during the time period:

We can rearrange the terms to determine the change in population abundance AN:

If we assume the population is closed, that is, there is no movement of individuals into the population (immigration) or movement out of the population (emigration), then the equation becomes simplified. In this case, we are modeling only the changes that occur within a population. Although most populations are not closed (i.e., there is immigration and emigration), it allows for some easier calculations and allows us to provide details on a specific population. For a closed population, with I = 0 and E = 0, the equation simplifies to

If we also assume that the period of time between estimates of population abundance is extremely small (i.e., t is nearly 0), then we can treat population growth as continuous. If we estimated population abundance only after a large period of time then population growth would not be modeled as continuous but would be a discrete function. Modeled as a discrete function the estimate of the population abundance would be the same or changing during the time period, then at the end of the period it would change to a new level, stepping from one abundance level to another after each time period (Figure 2). By assuming population growth is continuous we can use a differential equation to describe the change in population size (dN) that occurs during an extremely small period of time (dt):

B and D now represent rates, or the number of births or deaths in this population during these short periods of time. The birth rate (B) can be thought of as the number of births per individual per time period (b), called the instantaneous birth rate, times the number of individuals in the population (N) at that time. The death rate (D) also can be calculated using the instantaneous death rate (d) times the population abundance (N). The continuous differential equation is now

Continuous, exponential (+r)

Discrete (+r)

Figure 2 Various models of population change: a continuous exponential population increase (when r is positive), a continuous exponential decrease (when r is negative), and a discrete model (when r is positive).

This is very much a simplification of the real world (a model) because it assumes that the instantaneous birth rate and death rate are constants, thus per capita birth and death rates are the same no matter what the population abundance. We know this is not true, because for many populations the birthrate will decrease and the death rate increase when populations become larger. Thus the factors affecting population growth are dependent on the population abundance or density, hence are called density-dependent factors. The concept of density-dependent factors will be discussed later.

The instantaneous rate of increase of the population (r), often called the intrinsic rate of increase, is b — d, so substituting r for (b — d) produces a new model for population growth that is dN

This simple equation allows us to predict that the change in population abundance per unit time is exponential and proportional to a constant value of r (Figure 2). If the intrinsic rate of increase is zero then the reproductive rate and mortality rate are equal (b — d = 0). If r is positive or greater than 1 (per capita birth rate exceeds per capita death rate) then the population would increase proportional to the population abundance (N) in an exponential fashion. If r is negative or less than 1 (death rate exceeds the birth rate), then the population would decrease exponentially (Figure 2). The greater the value of r the more rapidly the population will change. The intrinsic rate of increase can be used to model or predict future population abundance (Nt +1) by integrating the population growth model and knowing the time interval (t) over which you want to predict abundance and the initial population abundance (Nt):

This simple exponential model has many assumptions. We assumed the population is closed, a simplification to eliminate the effects of immigration and emigration. This assumption will be removed during a discussion of metapopulation models. We assumed that per capita birth and death rates were constant, but they do change with changes in population size, and with different age and sex classes. We also assumed that population growth was continuous; however, many species have discrete generations, where births and deaths are measured not instantaneously but at certain longer time frames. Population dynamics of species that reproduce once per year, for instance, often are modeled not with a continuous but with a discrete model. Finally, this model is deterministic; it will always predict the same abundance given an initial population size and a specific r and t. Adding variability to the model produces a stochastic model (i.e., the effects of random variation that may be more realistic), and predicted abundances are affected by the variability in the intrinsic rate of increase. If r is positive but has a great deal of variability, the population will increase and the predicted abundances will have greater variability through time. The consequence is that through random chance, the population could go extinct because of an extreme negative random effect although the population was increasing previously.

As a population begins, growth rate is minimal because there are few individuals and many are probably young and not producing offspring. After the early stages, growth becomes more rapid, often increasing exponentially, because there are now more individuals that are reproducing and factors that may restrict population growth, such as food resources, space, predators, diseases, are not affecting the population in a large manner. During this phase as the exponential model would predict, population growth is proportional to the population size and magnitude of r. Eventually, as the population increases in size, nutrients become more difficult to locate or obtain, predators or herbivores are more abundant, competition among individuals within the population increases, mortality, because of disease, parasites, and movements into suboptimal areas increases, and space or shelter becomes limiting. These factors to some extent control the ultimate size of the population, causing density-dependent regulation. The maximum size that the population can achieve is often referred to as the carrying capacity (labeled K). Carrying capacity is not really constant nor is it the maximum capacity but the population often fluctuates through time as various conditions within and outside the population change. The carrying capacity is more realistically determined by examining the long-term average of abundance. Species that live in relatively stable environments (e.g., invertebrates in the deep sea) may have relatively constant population numbers around the carrying capacity, whereas those populations in unstable environments (e.g., nearshore invertebrates) may have variable numbers through time. Large regime shifts or major changes in environmental conditions caused by natural or anthropogenic factors can produce a shift in the carrying capacity for the population.

Because organisms cannot increase exponentially forever, the concept of a carrying capacity (K) was developed for the population growth models. The simplest relationship is to assume that changes in per capita birth rate (b) or death rate (d) are linearly related to population size. As population size increases b decreases linearly and d increases linearly. The concept makes sense; as the population increases, reduced resources, competition, predation, and other density-dependent factors would cause an increase in deaths and decrease in births. It probably is intuitive also that the relationship between b and d with population size is not linear, but more likely some nonlinear function, with the changes in b and d becoming more pronounced as density becomes extreme. Carrying capacity occurs when the population abundance reaches a maximum equilibrium (N = K), when birth rate and death rate are equal. The population model that depicts the effect of carrying capacity on population growth is the logistic model:

This is basically the exponential growth model with another term (1 — N/K), that is, the portion of the carrying capacity that can still be filled with the population. When the population is in its infancy (N is extremely small compared with K), then N/K is small and (1 — N/K) is nearly equal to 1. At this early stage the population grows exponentially (rN). As the population increases and approaches K, the ratio of N/K approaches the value of one, (1 — N/K) approaches zero, and the change in population size (dN/dt) approaches zero (Figure 3). Thus as the population approaches K the rate of change in the population slows or decreases, effectively stopping exponential positive growth (Figure 4). If the population should exceed K, then (1 — N/K) is negative and the rate of change will be negative and the population will decrease, presumably toward K. In the logistic model, the greatest population growth rate is achieved at half the carrying capacity (K/2), and populations with a greater value of r will reach K more rapidly (in less time) than populations with lesser values of r.

Because the various density-dependent factors do not affect the population growth rate immediately, often there is a time lag between the change in population size and effects on population growth. This lag time effectively Figure 3 The change in population per unit time (dN/dt) relative to population size (N) for a population increasing exponentially and one using a logistic model. Figure 4 The logistic population model plotting the population size (N) through time.

Time

Figure 4 The logistic population model plotting the population size (N) through time.

creates an oscillation of the population abundance around carry capacity, because the population adjustment by the carrying capacity does not take effect immediately; so abundance will go past K, and then dip below K If the combined effects of lag time and response time (1/r) are small then the population will increase in a smooth logistic fashion; if the effects are moderate, the population will oscillate around K, eventually decreasing in oscillation until the population is at K. If the combined effects of lag time and r are large, the population abundance will oscillate around K without reaching an equilibrium value. The amplitude of the oscillation around K will be greater if the population grows rapidly (r is great) and the time lag is great. Many populations oscillate around K because the effects of reaching carrying capacity, such as increased number of predators, decreased reproductive output, and increased aggression, take some time before they have an effect (there is a substantial lag time). With greater levels of r, population dynamics can become chaotic, in the sense that there is no stable equilibrium and no stable cycles. In a chaotic system you cannot predict the cycle, but the population could still fluctuate around a long-term mean.

The previous models best approximate population growth rates for smaller organisms (e.g., bacteria), where the organisms reproduce rapidly (r is great) and have short generation times. Many larger organisms, however, can be modeled using age- or stage-based models often called life table models (Table 1). In these models, probability of survival to the next age class (1x) and mean number of female offspring per female of an age class (mx) are computed for each age class and used to compute net reproductive rate (Ro):

Table 1 A theoretical life table, perhaps of a mammal, because mortality is greatest in the early years, less during midlife, and increases again near maximum age

No. alive at Proportion No. dying within Finite Mean no. of female beginning of surviving until age interval rate of offspring produced

Age, x age interval, beginning of age (from x to x + 1), Finite rate of survival, within age interval,

(years) nx interval, lx dx mortality, qx px mx lxmx

 1 1000 1 480 0.480 0.520 0 2 520 0.52 300 0.577 0.423 0 3 220 0.22 120 0.545 0.454 0 4 100 0.1 15 0.150 0.850 0.5 5 85 0.085 23 0.271 0.729 1.5 6 62 0.062 21 0.339 0.661 2.2 7 41 0.041 13 0.317 0.683 3.3 8 28 0.028 11 0.393 0.607 3.1 9 17 0.017 10 0.588 0.607 2.8 10 7 0.007 7 1.000 0.000 1.8 11 0 0 In this theoretical case, the population would be expected to be decreasing because Ro is less than one (Ro = 0.597). Ro is the potential number of female offspring produced by a female during her lifetime. If Ro is greater than 1 then the population will increase because a single female is producing more offspring than she needs to replace herself.If Ro is less than 1 then the population will decrease, and if Ro is one the population is stable. Generation time is the average age difference between a cohort (a group of individuals born at the same time) and their parents or the amount of time it takes a cohort to replace another cohort. Generation time is calculated as Ei=0 lxmxx Ex= 0 lxmx Using these age-based models, r is determined by solving in an iterative fashion, the following equation: The value of r can then be used in the logistic equation. The advantage of using age-based models is that the number of individuals in each age class can be estimated. Often the age-based life table is converted to an age-class matrix (a Leslie matrix) that is used to model changes in the age structure with population growth. In these models, with a constant birth and death rate, the population will attain a stable age distribution (i.e., relative numbers in each age class will remain equal as the population increases), no matter what the initial age structure. These previous models have depicted the changes in abundance for a single isolated population, but most populations are not isolated and are affected by the immigration and emigration of individuals (a open population). Certain population models have been determined that allow for open populations, and treat the system as a metapopulation (a group of populations that are linked by individuals that emigrate from one population to be immigrants in other populations). The long-term existence of the metapopulation is dependent on the probability of extinction of populations and probability of colonizing new spots (based on emigration rates). If there is a reasonable amount of emigration, even large probabilities of population extinctions usually predict the continued existence of the metapopulation because new recruits always are arriving to colonize new locations. The continuation of the metpopulation also is positively related to the emigration of individuals into populations that already exist, which increases numbers within existing populations, thereby decreasing the probability of population extinction. The theories associated with mark-recapture techniques are fairly well developed, allowing the observations of marked individuals in future sampling to be used to model immigration and emigration and population size. 