ecological dynamics and odes2. assumptions and hypothesis volterra developed a model based on a few...

Ecological Dynamics and ODEs

Dana Botesteanu and Matt Guay

Maryland Mathematical Modeling Contest

October 1st, 2013

Maryland Mathematical Modeling Contest Ecological Dynamics and ODEs

Ecological dynamics

Interactions of organisms in natural environments - prettycomplicated

To understand some facet of an ecosystem, we can usemathematical models

Objective: A good model is simple (abstracts away irrelevantdetail) but not too simple (captures phenomena of interest).

First step: Know what you want to model, and what youdon’t!


Predators and prey

Start simple! Build the simplest possible model beforeworrying about elaborations.

For us: One predator species, one prey.

Which aspects to model? Start with population sizes.

Populations changing over time could be modeled with..ODE’s!


A Predator Prey Model - Lotka-Volterra System


Lotka-Volterra Model

Structure

1 Historic origin & motivations;

2 Assumptions and hypothesis of the Lotka Volterra system;

3 Analytical solution;

4 Beyond Lotka Volterra - More realistic models.


1. Historic origin and motivations

Vito Volterra (1926) first proposed a model to explain the strangebehavior of the level of fish catches observed by sailors in theAdriatic Sea. Alfred J. Lotka developed the same modelindependently and simultaneously.


2. Assumptions and hypothesis

Volterra developed a model based on a few assumptions andobservations.

1 The prey in the absence of predation grows unbounded.

2 In the absence of prey, the predator population decreasesexponentially.

3 The predator effect is to reduce the prey growth rate,proportional to both the predator and prey populations.

4 The prey effect on the predator is to increase the predatorgrowth rate proportional to the prey and predator populations.


3. The Mathematical Model. Analytical Solution


4. Beyond Lotka Volterra - More realistic models


Other modeling approaches

We can revisit the way we model predator-prey interactions.

ODE models look at dynamics in a low-dimensional statespace (in this case, R×R)

Idea: Let the state space correspond to physical space (sorta).

States are objects in physical space (sorta)

Time is discrete

Simplest approach here: cellular automata (CA).


Cellular automata

Underlying state space G is usually a grid of M “cells”{xi}Mi=1 (though other graphs work, too).

Time variable t ∈ N (usually)

The possible states are a small, finite set S, e.g. S = {0, 1},S = {red, blue, green}, S = {fox, rabbit}.We still just have a function on this domain in our model,ut(x) ∈ S ∀ t ∈ N, x ∈ GInstead of derivatives giving information on functionschanging in time, use transition maps that specify how a cell’sstate changes from one time step to the next.


Cellular automata

Transition maps: in general,

ut+1(x) = F ({uk(y)}, t) ∀ k ∈ [0, t], y ∈ G,where the transition map F is a function (deterministic CA)or a stochastic process (stochastic CA) taking values in S.Discrete dynamical system: ut+1(x) = F ({ut(y)}), i.e. onlydepends on G’s states at the previous time step.Commonly, ut+1(x) depends only on ut(x) and ut(y) for y ina neighborhood N(x) of x.

Figure: Two common neighborhoods; (a) Von Neumann and (b) Moore.


A predator-prey CA model

Assumptions:

1. Two “animals”: fox (predator) and rabbit (prey).

2. Foxes can move, die, eat, and reproduce with someprobabilities.

3. Rabbits can move, die, and reproduce with some probabilities.

Setup:

State space is an N ×N grid.

States are empty (E), fox, (F ), rabbit (R).

The transition map is stochastic and best describedalgorithmically, using Moore neighborhoods.


The transition map

Adapted from Hawick & Scogings, 2010

For each time step t+ 1For each cell x (chosen in random order)Choose y in neighborhood N(x) at randomIf ut(x) = F and ut(y) = Rut+1(x) = E, ut+1(y) = F with probability εf (fox eats rabbit)

Else if ut(x) = R and ut(y) = Fut+1(x) = F , ut+1(y) = E with probability εr (rabbit eaten by fox)

Else if ut(x) = F [R] and ut(y) = Eut+1(x) = E with probability δf [δr] (die)ut+1(x) = F [R], ut+1(y) = F [R] with probability ρf [ρr] (reproduce)ut+1(x) = E, ut+1(y) = F [R] with probability µf [µr]. (move)


A predator-prey CA model

Things to note

The mathematical formalism is instructive in general, but analgorithmic description is more useful for most models.

Lots of parameters! εf , εr for eating, δf , δr for dying, ρf , ρrfor reproduction, µf , µr for moving.

Parameter values dictate system dynamics. Extinction of oneor both species, or cyclic population growth and decline (a laLotka-Volterra) are all possible.


(Start simulation now)


Model strengths and weaknesses

Strengths:

Captures more aspects of population dynamics thanLotka-Volterra.

CA’s allow simple, well-chosen rules to generate complexbehaviors.

Easy to program.

Large numbers of parameters mean behavior can be tailoredto known data.

Easy to modify for better model fidelity.



Weaknesses:

Still fails to capture many aspects of predator-prey dynamics.

High-dimensional state spaces mean analytic results aredifficult to produce.

Simulation via cellular automata is usually inductive ratherthan deductive.

May be computationally intractable for large domains.


Elaborations on this model

Instead of a grid, consider an automaton on a general graph,for better spatial fidelity.

Create a more complicated food web by adding additionalpossible CA states.

Investigate agent-based models instead of CA models.

Rules can be made to vary in space and/or time.


And now...

Contest Tips 1


Contest necessities

Everyone should have a computer to work on.

Look for a (reasonably) comfortable working space ahead oftime.

Software to write up your solution (LaTeX)

A programming language at least one (preferably two)teammates can use.

Be able to learn, quickly!


Suggested timeline

Before contest begins: Coordinate! Know where you’llmeet, exchange email addresses and phone numbers. Knowwhen teammates won’t be available

Friday: Problem is put online at 5PM. Time for research. Doas much background research on the problem as you can.Start outlining at least two possible modeling approaches.

Saturday: Keep doing background research. Choose amodeling approach, start programming an implementation.Start writing. Suggested: 2 working on the model, 1 writing.

Sunday: Both implementation and writing should be in fullswing. By Sunday night, 2 people should be writing. Don’t goto sleep.

Monday: Solution is due at 10AM sharp. Plan to finish by9AM.


Solution paper structure

Abstract

Title page, table of contents

Problem description

Model description (including proposed solution)

Model assumptions

Results


Conclusion

Code appendix

Works cited


Finding and using documents

Obvious starting places: Google, Google Scholar. Researchpapers > random websites.

http://www.lib.umd.edu/ may have access to papers youcan’t get on Google Scholar.

Investigate references in papers you’ve already found.

Google Scholar also lets you see who has cited a given paper(super helpful).

Keep a running bibliography, even of papers you aren’t sureyou’ll use. You can trim it at the end.


http://www.lib.umd.edu/

Finding and using software

You’ll likely need new software, or software libraries during thecompetition.

Use existing code when possible. Don’t write your own unlessyou have to!

Finding and using new software/code means knowing how tosearch effectively.

Look for documentation or help pages.


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