Modelling life lecture 1 van Doorn
Modelling, change and simulation
When we see such a pattern as biologist, we can try to make sense or describe this pattern in
two ways. We can focus on the pattern itself (are the oscillations regular, do they show a
fixed amplitude etc.). We can use all kinds of tools for this. This relies strongly on data
analysis and statistics. A second way is to focus on (biological) cause, so what the drive is
behind the oscillations. So, what are the underlying mechanisms that lead to these
oscillations. This is what we focus on.
For this example: there are multiple potential mechanisms. One is that the oscillations are
driven by fluctuations in food availability. This may have to do with external factors as solar
sports or El Nino events. It may also have to do with social dynamics (rises in population
density lead to differences in behaviour which lead to differences in growth rate of the
population which might lead to a decrease. Another explanation is that the cycles are driven
by predation.
The idea is that lynx and hare dynamically affect each other’s population density by
ecological feedback. The figure is the basic idea that we try to capture in the mathematical
model. Let’s do this. To indicate that we switch to a mathematical model, we switch to
another model (sharks).
Prey = tuna, predator = shark. To derive a mathematical model for the interaction with these
two species, we start with a description that details how the change in the density of these
two animals depends on ecological processes. We use mathematical notation.
Shark density at time t is noted as S(t).
Tuna density at time t is noted as T(t).
Then we derive our model by saying that the change in S or T depends on the amount of
shark births and deaths and tuna births/deaths. Those elementary biological processes we
must make explicit. How the tuna depends on the shark and the other way around.
Shark births are depended on the shark density, but we also assume that the rate of the
shark births depend on the tuna density. If there is more tuna, the shark has more food and
can produce more offspring. Shark deaths are solely depended on shark density (more
sharks = more deaths), it is independent of the tuna density.
For the tuna we must make similar assumptions. Tuna births is depended on the tuna
population density. Independent of how many sharks there are, tuna will reproduce at a
certain rate. Tuna deaths are depended on the tuna density but also the shark density.
We assume that sharks have d, per-capita death rate of sharks. This captures the death rate
of the shark population. So, every shark has a constant rate of death. So, if we multiply d *
S(t) we have the number of sharks that die at a certain time due to normal causes.
,b is the per-capita birth rate of tuna. We assume that the birth rate of tuna is constant, so
every tuna produces an offspring at a constant rate. So, b is the per capita birth rate of tuna.
So, the entire population produces offspring at a rate of b * T(t).
One assumption is that we say that shark and tuna are coupled by an ecological interaction.
One simplified assumption is that this is the only death cause of tuna (that sharks eat them).
then we say that the number of deaths of tuna depends on the rate of sharks and tuna
meeting. So, this depends on the density and when they meet. So, 2 times more sharks
mean 2 times more encounters. So, the meet rate is proportionate to S and T. So, the death
of the tuna can be explained by – a*S(t)*T(t). The attack rate times the amount of sharks
times the amount of tuna. The attack rate means the probability that the meeting of a shark
and a tuna will lead to tuna death.
The inflow of S(t) depends on the amount of tuna, because more food means for offspring.
So, for this part, you need to multiply the attack rate with the tuna-to-shark conversion
factor and the shark and tuna densities.
These models establish a relationship in a current state of the variables (shark and tuna
densities) and the change of the variables.
S’ en T’ is the change in S or T in a unit of time. So ‘ is the change in the sort in a unit of time.
This change is also called a differential equation which describes how the rate of change of a
state variable depends on time and/or on the value of the variable at the current time point.
A system of differential equations consists of relationships for multiple state variables: each
variable’s rate of change may depend on all other variables (i.e., equations may be coupled)
and may also depend on time directly.
The shark tuna model is a system of differential equations, where none of
the right-hand sides depend explicitly on time t. such systems are called autonomous.
Typical steps for the equations:
1. What info is necessary to specify the current state of the system define the state
variables.
2. Specify the inflows and outflows for each state variable. What processes are
responsible for each of them changing? E.g., birth and death.
3. Determine structural dependencies. Which of the inflows and outflows are
influenced by which of the variables? E.g., the birth of the tuna is unaffected by shark
density.
4. Specify quantitative relationships for each of the flows. So, how do the current values
of the variables determine the rate of each process?
What can we do with this model? We can analyze it.
If we know the values of the parameters and we know the current population density, then
we can evaluate the rate of change of the variables to know how the populations will be
changing from the current time point onwards.
, Once we have the values of the variables, we can evaluate the right-hand side of
the equations. We can see that the change in the shark population is positive, this
means that the sharks are on the increase. For tuna’s it is negative, they are on the
decrease. We can plot this. Arrow for tuna is to the left, for sharks it goes up. The
direction of change can be represented graphically by an arrow in state space. If
we follow the same procedure for many different starting points (T, S), we obtain
a vector field. This visualizes the direction and rate of change for a differential
equation model for many different combinations of the state variables. If we
follow the arrows of the vector field, this gives us a way to reconstruct how the variables
change through time.
Simulating change
For this model, you can ask if you can determine how far you have travelled at time t. one
way to do this is to approximate your position. So, if X(t) is your position at time t, then your
position at a time point a little bit later in time is given by: X (t + Δt) = X(t) + Δt * v(t). We can
also write this differently, X (t + Δt) = X(t) + Δt * X’(t).
So, we are assuming that the rate of change (velocity) will be constant for Δt. But this is not
always the case, so we make an error. Therefore, it is an approximate solution.
We can also apply this to other models, also the shark and tuna model. We take an initial
point (T0, S0) at time t = 0 and we evaluate the rate of change at this point according to the
differential equations. We know the current states of T and S, this gives us values for the rate
of change.
This method is called the Euler integration method. It constructs a solution to the differential
equation model following the arrows of the vector field. This produces an
approximate solution. The rates of change do not really remain constant during
time interval Δt. The accuracy can be improved by reducing stepsize Δt but this
requires more computational work. So, starting from an initial condition we can
now proceed in tiny steps, following the direction of the vector field, to
reconstruct how the state variables change over time. This approach yields
trajectories in state space.
There is a relationship between trajectories and regular timeseries plots of the
variables. Solutions of differential equations are often represented in a phase
diagram but then we don’t have any information about time. The solution curve
that starts at a given initial conditions is unique and it corresponds to a unique
time series plot of the variables.
Book summary:
*Feedback systems can be positive or negative. These systems can respond to interventions
counterintuitive. So, if you for example remove almost all sharks of the population. Then
there will be a drastic increase in the tuna population and therefore there will be an even
higher shark population then there was before the intervention.
