Chapter 1: Exponential Population Growth

Model Presentation and Predictions

  • Population: a group of animals, plants, or other organisms of the same species that live together and reproduce

Variables for basic population growth models:

  • N: size of the population

  • t: time

  • \(N_t\): number of individuals in the population at time t

  • t = 0: can be considered the starting point of a population (or for calculations of population growth)

Note: The biological details of population growth vary tremendously among different speecies and can even vary among different populations within the same species.

Variables that cause changes in Population Size:

  1. Birth

  2. Death

  3. Immigration

  4. Emmigration

All of these factors work on different spatial scales as births and deaths depend on current population sizes while immigration and emmigration depend on the movement ability of individuals.

Basic Equation for Population Growth

The four factors covered previously can be combined into an expression for population growth:

\[N_{t+1}=N_t+B-D+I-E\]

To simplify the equation:

\[\Delta N = B - D + I - E\]

To get even more basic and consider an expression for a closed population (a population where there is no immigration or emmigration which is not really true in nature but mathematically allows us to focus on local population growth) we consider I and E to equal zero and do not consider them and thus:

\[\Delta N = B - D\]

To get a more realistic expression, the population growth will be modeled as a continuous differential expression:

\[\frac{dN}{dt} = B - D\]

Density independent population growth

Note: keep in mind that population size is important when thinking about birth and death rates as larger populations will probably produce more offspring (and subsequently have more deaths) than a smaller population over a short period of time.

We will assume each individual produces the same number of offspring to make things simple.

Since we are focusing on instantaneous rates with our continous expression, we will also have instantaneous birth (b) and death rates:

  • b = instantaneous birth rate [births/(individual*time)]

  • d = instantaneous death rate [death/(individual*time)]

Note: the instantaneous birth and death rate are measured per capita or per individual and to get the population birth rate you simply multiply by the population size

Note: We also assume with the below expressions that b and d are constant and dont change with population size which is not necessarily the case in nature as crowding can affect the birth and death rate

\[B = bN\] \[D = dN\]

However, this is not always the case as there may be time lags because current growth rate may depend on the population size at and earlier time

  • such as that with seeds that may remain dormant for a period of time

Exponential Population Growth Model

\[\frac{dN}{dt}=(b-d)N\]

b-d=r

\[\frac{dN}{dt}=rN\]

r is the instantaneous or intrinsic rate of increase. The value of r determines whether the population grows (r > 0), remains stable (r = 0), or declines to extinction (r < 0).

  • r measures the per capita rate of population growth over a short time period

  • Since r is proportional to N, the larger the population size, the faster it will grow

Question! When will \(\frac{dN}{dt}=0\)? - When N = 0 (assuming a closed population because migration would allow for an increase in population)

  • When r = 0 (meaning the per capita birth and death rates are equal)

Projecting Population Size

The previous equation tells us the population growth rate but no the population size at a given time interval. However, through integration of the expression we can get population size:

\[N_t = N_0e^{rt}\]

Exponential Growth Models

Note: Mathematically populations never reach less than one (this is considered extinction)

Calculating Doubling time

An important feature of population growth that is growing exponentially is doubling time. Regardless of the initial size of the population it will always double in size after a fixed period of time. Below is the derivation for the equation

First we know simply that N double is just 2N so:

\[N_{t_{double}} = 2N_0\]

Then if we substitute the above back into the equation for predicting population size:

\[2N_0=N_0e^{rt_{double}}\]

Divide by \(N_0\)

\[2 = e^{rt_{double}}\]

Take the natural log of both sides:

$$ln(2) = rt_{double}

Then solve for doubling time to give:

\[t_{double} = \frac{ln(2)}{r}\]

Note: The larger r is the shorter doubling time would be (this would be the case for things such as single celled organisms).

Model Assumptions

Going back to the equation for exponential population growth: \[\frac{dN}{dt}=rN\] there are major assumptions that we must consider when applying these mathematical formulas to biological examples. The predictions of these models depend on the underlying assumptions and if some assumptions are violated or the model is changed, then so will the results of the model as well. The major assumptions for the above model are as follows:

No I or E: The population is closed and changes in population size only depend on birth and death. This assumption allows us to focus on changes in a single population. Addition of immigration and emmigration would allows us to focus on multiple populations.

Constant b and d: We assume the population is growing with a constant birth and death rate. With this we also assume that there is an ultimited supply of resources avilable such as space, food, etc, as if we didn’t b and d would change as the population grew. We also assume that b and d don’t flucuate with time.

No genetic structure: We will assume that all individuals in a population have the same birth and death rates, therefore there is no underlying variation in the population that would affect these traits. However, if there is a genetic structure in the population it must be constant through time and in this case r is the average for the different genotypes in the population.

No age or size structure: We will assume there are no differences in b and d due to age or body size. This means that essentially the population is sexless and parthanogenic and can reproduce as soon as they are born (bacteria and protozoa most closely resemble this). If there are differences among ages, they must remain stable through time and r is an average across age classes.

Continuous growth with no time lags: Since the model is written as a differential equation births and deaths are occuring continuously and that r changes instantly as a function of N.

The most important assumption of this list is constant b and d which implies unlimited resources for population growth. This is because if the other assumptions are violated, the population will still increase exponentially (though a bit more complicated), but we need unlimited resources for the type of growth this model assumes.

Question! If unlimited resources aren’t actually a thing in nature why do we use this model?

  • All populations have a potential for exponential increase.

  • Exponential growth was a key factor in Darwin’s theory of natural selection as through his readings of Malthus’ writings he thought that the surplus of offspring resulting from exponential growth would cause more offspring than could survive and allow for natural selection to occur

  • Resources may be temporarily unlimited so populations can infact go through periods of exponential growth

Model Variations

Continuous versus Discrete Population Growth

In many cases time does not behave as a continuous variable. For example, in seasonal environments, many insects reproduce only once and then die and the offspring that survive are the basis for the population of the next year. Since birth and death rates are constant the population has non-overlapping generations and is thus modeled with a discrete difference equation.

Discrete Population Growth

If a population increases or decreases by a costant proportion \(r_d\)**(the discrete growth factor) we can start with the equation to model a population at timestep t+1 as: \[N_{t+1} = N_t + r_dN_t\]

Which is then simplified to:

\[N_{t+1} = N_t(1+r_d)\]

We will assume that 1 + \(r_d = \lambda\) the finite rate of increase so:

\[N_{t+1} = \lambda N_t\] Note: \(\lambda\) is always a positive number that measures the proportional change in a population from one year to the next. This means that it is the ratio of the population from the next time period to the current time period. It is also important to note that \(\lambda\) is a dimensionless number as it is a ratio of population sizes and because of this you can’t use simple scaling to change the time step.

The general equation for discrete population growth is as follows:

\[N_t = \lambda^tN_0\]

Note: The above equation is analogous to the equation we used previously to measure continuous population growth (\(N_t = N_0e^{rt}\)) and the continuous model is essentially just the discrete model with an infinitely small time step as it would look like a smooth curve as you reduce the timestep. Because of this we can switch between r and \(\lambda\) with the following:

\[r = ln(\lambda)\]

The relationship between r and \(\lambda\) are as follows:

r > 0 : \(\lambda\) > 1 (increase)

r = 0 : \(\lambda\) = 1 (stable)

r < 0 : 0 < \(\lambda\) < 1 (decrease)

Environmental Stochasticity

All the models we have previously covered are deterministic meaning that we will always get the same result if we put in the same number. The outcome is determined soley by the inputs and nothing is left to chance. Deterministic models are an idealized view of the world.

To understand stochastic models (models where there is variability), we need to understand that it incorporates two concepts mean and **varience(\(\sigma^2_x\)) which measured the uncertainty asssociated with the mean.

  • If varience is small, most random values will fall close to the mean
  • If varience is large, most random values can fall in a wide range from the mean

Environmental Stochasticity: Variability associated with environmental factors that influence growth and decline in population over different time scales

Note: The further in the future you try to predict the population size, the more uncertain the model becomes.

  • Since the varience of \(N_t\) is proportional to the mean and varience of r populations that are growing rapidly or have variably r flucuate more than slow growing populations or with more constant r.

  • If varience of r is zero than it becomes a deterministic model.

  • If N flucuates too rapidly the population may crash to zero.

Demographic Stochasticity

Demographic stochasticity occurs because most organisms reproduce in discrete units (integers). With demographic stochasticity the amount of births and deaths do not necessarily follow a deterministic order and can vary.

The equation for varience ofpopulation size if b and d are equal (which means the population will not increase on average):

\[\sigma^2_{N_t}=2N_0bt\]

Similar with environmental stochasticity, with increased time the varience in population size increases. There is still a risk of extinction even with populations with positive r values.

Note: Demographic stochasticity is especially important at small population sizes because it dosen’t take many sequential deaths to drive the population to extinction. The probability of extinction depends not only on the relative sizes of b and d but also the initial size of the population.

Note: Populations with high birth and death rates will be more variable than populations with low rates. The value of r may be the same between the two populations but the larger values will have a larger turnover.

Note: Average population size in stochastic models of exponential growth is the same as the deterministic model.