1. Computational Models

2. Calc I in One Slide…. Consider the function: We denote the derivative:
In this case:

3. Unconstrained Population Growth
r is the growth rate. P is the population at a given time.
This is the differential equation because it includes a derivative.
Solving for this differential equation means finding an equation for P given an initial population and a time.

4. Solving This one can be solved:
Unfortunately, it is usually impossible.
In those cases we approximate with a finite difference equation:
This looks like Python code!

5. Constrained Population Growth
It’s unrealistic to suppose that a population can grow exponentially forever.
Assume the system has some carrying capacity, call it K. (This is the max # of organisms that can be supported.) Then the differential equation/rate of change of the population looks like:

6. Lotka-Volterra Equations: Multiple Interacting Species
Track the population of two species:
V (prey)
P (predator)
Here, kv is the growth rate of V, kVP is the proportionality constant for the reduction of V interacting with P, kPV is the constant for the increase of P interacting with V, and kP is the death rate of P.

7. Analyzing Lotka-Volterra
This is getting complicated. What can we do to understand the system?

8. Analyzing Lotka-Volterra
This is getting complicated. What can we do to understand the system?
Solve for the equilibrium points!
(These are points where the derivative is always zero.)
We find:

9. Equilibrium Points A system may or may not have equilibrium points.
Three different kinds:
Unstable: system heads off to 0 or infinity if it is perturbed
Stable: system returns to the equilibrium point if it is perturbed
Marginally stable: system oscillates when perturbed
Activity: Look at Lotka-Volterra module

10. Differential Equations: They’re not just for Population Modeling!
An incredibly versatile tool:
Epidemic modeling
Modeling of physical systems
Planets
Pendulums
Connonballs
The list is endless.

11. Differential Equations: They’re not always the Right Tool
Some examples….
Empirical models (based on data, used to make predictions)
Simulations
Randomness
Cellular automaton

12. Cellular Automaton (CA)
Discrete model
Consists of grid (of any finite dimension) of cells, each in one of a finite # of states.
Each cell has a neighborhood consisting of a specific set of cells relative to it.
New generations are created based on a set of rules determining states of cells. Rules applied to each cell simultaneously.
Conway’s Game of Life!

13. Lattice Models Similar to CA
Except, cells are selected at random, and randomly interact with neighbors.
Example:
(Reproduction)
(Predation)
(Starvation)

14. Mean Field Approximation
Differential Equation Approximation of this lattice model:
This would hold if:
The environment were infinitely large.
Every individual could interact with every other individual regardless of location.

15. Equilibrium Points
These give rise to one non-trivial steady state (equilibrium pt):
These are marginally stable!

16. Wouldn’t This Make More Sense?
(Reproduction)
(Predation)
(Starvation)
Probably, but it has boring stable equilibria.