Modelling interaction – Mathematical Modelling – Mathematics – TU Delft


Hello! In this video I will show you how to construct a system of differential equations for interacting populations. A client wants an aquarium with two kinds
of fish: rainbowfish and gouramis. The client insists on a kind of gourami that
is rather aggressive and sometimes eats rainbowfish. We are going to construct a model to simulate
what happens. Let me say immediately, that the numbers we
are going to use here are made up. We do not really know what would happen in
such an aquarium. So this is a purely speculative model. One of the advantages of mathematical modelling
is that no fish will be harmed by it. First, we will consider what happens with
the separate species. OK, let’s put the rainbowfish in this virtual
aquarium. The model for the population by itself is
this differential equation. You see that the aquarium is large enough
for a healthy population of 100 rainbowfish. As the client does not sell any fish, there
is no harvesting term. This is the graph of the solution: the population
grows logistically from 20 to 100 fish. The gourami are modelled to do less well by
themselves. We assume that without small fish to prey
on, their birth rate is lower than their death rate: the net growth rate g is minus a quarter. So without prey, the number of gouramis would
decrease exponentially from 5 to zero. Now we put the 5 gourami in the tank with
the 20 rainbowfish. Here are the equations. The gouramis will eat the rainbowfish, so a negative term has to be included in the equation for P. The gourami population will benefit from eating
the rainbowfish, so a positive term should be added
to the equation for G. For these interaction terms, we start simple: both terms are modelled as a constant times the product of P and G. Maybe this does the trick, and we can always
do another modelling cycle when it doesn’t. So what could be appropriate values for alpha
and beta? First we consider the alpha for the rainbowfish. Let us go back to the balance equation: how
does P change during a time interval of length Delta t? The first term is the limited growth term
without interaction. The second term is the interaction term. It describes the number of rainbowfish that
are eaten between time t and time t plus Delta t. So, if you leave out the Delta t: alpha times
P times G is the number of rainbowfish eaten per day. Let us take alpha equal to 4%. This means that each gourami eats 4% of the
rainbowfish per day. If there are 25 rainbowfish in the tank, and one gourami, the gourami would eat 1 rainbowfish a day. If there are 25 rainbowfish, five gouramis
would eat five times as many rainbowfish a day. So that scales as you would expect. Now back to 1 gourami. If we take a hundred rainbowfish, the gourami
supposedly gets more aggressive with all these rainbowfish invading its territory, and the
rainbowfish will be easier to catch, so the gourami now eats 4 rainbowfish per day. You see that when there are very many rainbowfish,
the model does not hold up: when there are 1000 rainbowfish, the poor gourami would have to eat 40 rainbowfish a day. So the conclusion is that when P is high this
model has to be improved. The capacity for our tank is only a hundred
rainbowfish, so we are probably safe. Now the equation for the gouramis. Here also we go back to the difference equation. The interaction term now describes the extra
growth in the number of gouramis, now that they have extra food. This growth could be caused by additional
eggs being hatched, or you could say that, because the gouramis have eaten the rainbowfish,
they die later. We set the value of beta at 0.008. You can interpret that as follows. Each gourami eats 4% of the rainbowfish each
day, and for every 5 rainbowfish that a gourami eats, one extra gourami is gained: 4% divided
by 5 equals 0.008. The two differential equations form a system. The equations are coupled: in the differential
equation for P, there is a term with G and vice versa. So the equations have to be solved together
as a system. In this video you have seen a simple predator-prey model, and how the interaction terms in it can be interpreted. In the course you will learn how to approximate
the solutions of systems numerically, and to determine the type of the equilibrium points.