## Population Ecology I

December 2, 2020 by anon

A major component of modern ecology is the study of different species in terms of their interactions, growth, and evolution. Population ecology is a vast and fascinating field that helps us quantify some of the changes that occur in an ecosystem. It enables us to observe patterns in the interaction and growth of different species in an environment.

If you were to model something like the human population or the population of a species in a particular environment, a wide variety of factors need to be taken into consideration.

However, to better understand the process, let us begin from the simplest case possible and build from there.

Consider bacteria that multiply in number by means of asexual reproduction.

Initially, say there are N0 bacteria in a Petri dish; they undergo mitosis and over a span of time, double in number. They then further divide, continuing to increase in number with each generation.

Let us attempt to model this mathematically.

Let N(t) be the number of bacteria at an instant of time t. The rate at which the population changes, i.e. changes to N with respect to time can be represented by the differential equation

$\frac{dN}{dt} = rN$

Where,

$r$ represents the growth rate and

$N(t)$ represents the population of bacteria at time $t$.

It can be observed that the term of the right-hand side is positive. This indicates that the population increases as time passes, and this increase is proportional to the growth rate and the population of bacteria present at that instant of time t. Essentially, the higher the growth rate, the faster the population increases; further, the higher the number of bacteria present at that moment, the more the bacteria that undergoes mitosis, and consequentially, the higher the population.

The above differential equation can be solved by method of separation of variables. This yields the following solution:

$N(t) = N0 \times exp(rt)$

From this expression, it can be observed that the growth is exponential and proportional to the initial population $N0$

Figure 1 illustrates a simple plot of how the population changes with time

Fig 1

However, this model has some very evident shortcomings. Any population cannot continue to grow indefinitely. This would require infinite resources, which is neither practical nor sustainable.

To overcome this, we can modify the model such that the population starts to decrease on reaching a specific value.

Consider the modified equation

$\frac{dN}{dt} = r N \frac{K - N}{K}$

Here, the variable $K$ represents the carrying capacity, a factor that provides the upper limit for the population size that can be sustained by the model. $\frac{K - N}{K}$ represents the fraction of utilization of the carrying capacity. Notice that growth rate reduces as the value of $N$ approaches that of $K$.

Thus, initially, the population grows exponentially, but it levels out as it approaches the carrying capacity K.

Figure 2 shows a simple plot of the population change with time according to this modified model

Fig 2

The above modified model is called a logistic growth model. It is useful in modelling population growth in simple cases, such as bacteria in a Petri dish, or yeast cells in fermentation bowls.

For example, in the case of a Petri dish, where you have an initial number of micro-organisms present and finite resources for nutrition, the initial population and the rate of growth will determine future populations. Further, the finite resources will determine the carrying capacity of the system. Thus, the logistic model can effectively predict the behavior of the population.

However, note the simplicity of the scenario we have considered. Here, there is only a single species present in an ecological niche. The logistic model does not account for any interaction between species. For example, if the species modeled is competing with another species for the same resources, or if the species serves as prey for another species, the population dynamics will be very different.

These shortcomings are handled in more complex models of population dynamics, like the competitive Lotka-Volterra model and the predator-prey model, which will be the subject of future articles in this series.

(all the code used for these models can be found on GitHub : https://github.com/desmondquinn/math-in-ecology)

Guest post written by Desmond Joseph Quinn

Desmond has always been fascinated by science and technology. He loves learning about the universe and all its little and not-so-little details. He is currently pursuing his master’s degree in Nanoscience and technology. Though his background is in physics, he like exploring beyond. He hopes to get you excited about some science through his writing.