When things have the right balance and are in the right proportions, the world is known to be in equilibrium. Such is the case with populations of different species coexisting in an environment. This project will consider how populations of foxes, rabbits and carrots interact with each other and how their population sizes vary over time on an isolated island. Under ideal conditions, the population of all species should grow and decline steadily, in a cyclical manner. Of course, nature does not always behave ideally; natural disasters occur and humans also inevitably interfere with the balance in nature. Can a model be created that most accurately represents a real-life situation and that incorporates realistic disruptions in real time? How might human interferences affect the balance of these populations? This project will feature a real-time graph that demonstrates the changes in three species’ populations over a century while incorporating the effect of two human disruptions that will be later discussed.
The project relies on the Lotka-Volterra Equations, and modifies them to make our model
more representative of real life. These equations are the most common way to describe and
model the basic ecological predator-prey relationship. Lotka-Volterra equations state that in
order for the model to be accurate, it must be representative of a closed ecosystem where no
migration occurs, like on our island. Next, there may only be two animals: predators and prey, in
a very simple food chain: the predator consumes prey, and the prey consumes some kind of
vegetation (which has an endless supply). The following 4 variables must be considered:
Using these variables, we can find the non-linear differential equations of each species:
- Prey, x: dx/ dt = Ax - Bxy → dx = Ax dt - Bxy dt
- Predator, y: dy/ dt = Dxy - Cy → dy = Dxy dt - Cy dt
As said in the introduction, we’d like to make it as realistic as possible: in order to make the populations’ growth more representative of real life, we can implement logistic growth instead of exponential growth. In exponential growth, the growth rate allows the population to growth endlessly, growing quicker as the population increases. In logistic growth, the carrying capacity, or maximum size of a population in certain conditions, restricts the size of the population, and the growth rate decreases as the population grows (essentially, the growth rate depends on the size of the population). Having a population ceiling is true in real life, so it should be incorporated into the differential equations. To get things started, the basic differential equation for population at a given time will look something like:
- dP/dt = R*P
Where dP/dt is the growth rate of a population at a given time, P is population size, t is time and R is the rate of increase per capita. If R is always at its maximum, Rmax, then the growth will be limitless, turning into exponential growth:
- dP/dt =(Rmax)*P
Logistic growth, as explained above will alter the differential equation for population growth:
- dP/dt = Rmax*((K-P)/K)*P
Where dP/dt is the growth rate of a population at a given time, P is population size, t is time, Rmax is the maximum per capita rate of increase and K is the maximum population, or carrying capacity. As the population approaches the carrying capacity, dP/dt heads towards zero. This should produce an a curve that closely resembles periodic motion. The improved differential equations using the same variables as before, K as the carrying capacity of the prey and Q as the carrying capacity of the predators now looks like:
- Prey, x: dx/ dt = A*((K-x)/K)*x - Bxy
- Predator, y: dy/ dt = Dxy - C*((Q-y)/Q)*y
Of course, the previous models, involving only two populations were more simple. New, more complex differential equations will be needed to try to model a predator/prey model with 3 species. In a three-species system, it is important to note that it is still an isolated environment and a simple feeding chain, where the carnivore consumes the herbivore while the herbivore consumes the vegetation. Our island features Foxes (carnivores), Rabbits (herbivores) and Carrots (vegetation). The food chain will look like:
- Foxes → Rabbits → Carrots
The resulting differential equations are the following:
As we can observe from the plot obtained from the code, the populations are balanced and reach a dynamic equilibrium: rising and declining in a cyclical manner. In fact, the graphs shows that the period of each cycle is approximately 10 years. The graph also demonstrates that the size of each population affects each other: the number of foxes depends on the number of rabbits, the number of rabbits depends on the number of carrots as well as the number of foxes, and the number of carrots depends on the number of rabbits.
As we take a closer look, we can notice that the population of the rabbits closely follows
the population of the carrots: as the number of carrots rise, the rabbits have more food and
therefore can grow more. The same is true for the foxes, as their population relies on the number
of rabbits, but to a slightly less extent.
When the human disruption is first introduced, at year 50, the chemicals in the soil eliminates the carrots. This disruption lowers the population of carrots quite significantly, which has a drastic effect on the population of rabbits, as the carrots are the only food source available for the preys. We can observe the population of rabbits decrease to its second lowest point in the simulation. The population of the foxes is also affected, but to a lesser extent as they are less dependant on the number of rabbits. Following this decrease, the populations come back to the dynamic equilibrium and continue to cycle with the same 10 year period as they did previously. When the second human disruption is introduced, at year 80, the population of foxes drop at a very rapid rate. This has some interesting effects; allowing the rabbits to grow, which prevents the population of the carrots to rise (as they normally would have in the cycle). After this, the excess of rabbits allows the population of foxes to grow relatively quickly, as the whole equilibrium becomes balanced again by the end of the simulation.
As shown, human interferences do have significant effects on the cycle of the populations residing on the island. In fact, when the simulation was run with disruptions that were too lethal to a certain species, that species’ population went to zero, and the balance was completely ruined. However, in a perfectly balanced ecosystem with manageable interferences, nature balances everything out and the populations will reach an equilibrium once these disruptions are over.