Scientific Computing I Part II: Population Models Module 2: - PowerPoint PPT Presentation

Scientific Computing I Michael Bader Outlines Part I: Fibonacci’s Rabbits Scientific Computing I Part II: Population Models Module 2: Population Modelling – Discrete Models Michael Bader Lehrstuhl Informatik V Winter 2005/2006

Scientific Part I: Fibonacci’s Rabbits Computing I Michael Bader Outlines Part I: Fibonacci’s Rabbits Part II: Population Models A classical population model 1 Looking for an improved model 2

Scientific Part II: Population Models Computing I Michael Bader Outlines Part I: Fibonacci’s Rabbits Discrete vs. Continuous 3 Part II: Population Models Deterministic vs. Stochastic 4 Spatial and Temporal Resolution 5 Single- vs. Multi-Population 6 Level of Detail 7 Finally: What’s the Task? 8

Scientific Computing I Michael Bader A classical population model Looking for an improved model Part I Fibonacci’s Rabbits

Scientific Fibonacci’s Rabbits Computing I Michael Bader A classical population model Looking for an improved model A pair of rabbits are put in a field. If rabbits take a month to become mature and then produce a new pair every month, how many pairs will there be in twelve months time? Leonardo Pisano (“Fibonacci”), A.D. 1202

Scientific Model Assumptions Computing I Michael Bader A classical population model Which assumptions or simplifications have been Looking for an made? improved model we consider pairs of rabbits rabbits reproduce exactly once a month female rabbits always give birth to a pair of rabbits newborn rabbits require one month to become mature rabbits don’t die . . . ?

Scientific The Fibonacci Numbers Computing I Michael Bader How many pairs of rabbits are there? A classical we start with a newborn pair of rabbits population model Looking for an after one month: still 1 pair of rabbits (now improved model mature) after two months: 2 pairs of rabbits (one mature) after three months: 3 pairs of rabbits (two mature) after four months: 5 pairs of rabbits (three mature) after n months: f n = f n − 1 + f n − 2 , f 0 = f 1 = 1

Scientific The Fibonacci Numbers (2) Computing I Michael Bader A classical Now: how many pairs of rabbits are there? population model Looking for an f 10 = 55, f 12 = 144, f 18 = 2584, . . . improved model exponential growth of rabbits: f n = 1 ( φ n − ( 1 − φ ) n ) , √ 5 √ � � where φ = 1 1 + 5 ≈ 1 . 61 . . . is the golden 2 section number. questions: how accurate is the model? what are its shortcomings?

Scientific Wanted: An improved model Computing I Michael Bader Group Work: A classical population model Develop an improved model for the Looking for an improved model growth of a rabbit population! Model assumptions: 1 what assumptions do you want to keep what assumptions do you want to drop or modify Describe your model 2 Describe how to run the simulation 3 starting conditions evolution of the population . . .

Scientific Comparison of models Computing I Michael Bader A classical population model Discussion: Looking for an improved model What are the differences between the proposed models? Consider: the modelling of the rabbits the interaction between rabbits the environment (time and space) possible external influences

Scientific Computing I Michael Bader Discrete vs. Continuous Deterministic vs. Stochastic Part II Spatial and Temporal Resolution Classification of Models Single- vs. Multi-Population Level of Detail Finally: What’s the Task?

Scientific Discrete vs. Continuous Models Computing I Michael Bader Discrete vs. Discrete Population Modeling: Continuous Deterministic vs. count individual rabbits (pairs of rabbits) Stochastic Spatial and “clocked” evolution of the population: Temporal Resolution changes occur at discrete points in time or Single- vs. within time intervalls Multi-Population Level of Detail Continuous Population Modeling: Finally: What’s the Task? population size ∈ R continuous growth or decay ⇒ population size is a function: p : R → R , p ( x ) = . . .

Scientific Deterministic vs. Stochastic Models Computing I Michael Bader Discrete vs. Continuous Deterministic Population Modeling: Deterministic vs. Stochastic fixed birth rate, fixed gender distribution Spatial and Temporal model leads to uniform simulation results Resolution Single- vs. Multi-Population Stochastic Population Modeling: Level of Detail probability distribution for birth rate and Finally: What’s the Task? gender simulations may lead to different results; both, expected value and aberrations, may be of interest

Scientific Spatial and Temporal Resolution Computing I Michael Bader Discrete vs. Spatial resolution, only: Continuous population does not grow or decay Deterministic vs. Stochastic expanding and spreading of interest Spatial and Temporal Resolution Temporal resolution, only: Single- vs. Multi-Population growth and/or decay are of interest Level of Detail uniform population distribution in a fixed Finally: What’s the Task? region Temporal and spatial resolution how does growth/decay affect population distribution?

Scientific Single- vs. Multi-Population Models Computing I Michael Bader Discrete vs. Single population model: Continuous Deterministic vs. population of rabbits Stochastic no other species, but distinction between Spatial and Temporal male/female, healthy/ill, hungry/well-fed, . . . ? Resolution Single- vs. Multi-Population Multi-population: Level of Detail Example: rabbit population Finally: What’s the Task? competitors: everything that eats carrots!? predators: fox, man, . . . prey: carrots ⇒ Systems of interacting populations

Scientific Level of Detail Computing I Michael Bader Rabbit modelling: Discrete vs. Continuous “pair of rabbits” (mature/non-mature) vs. Deterministic vs. Stochastic male/female, x years old, healthy/ill, Spatial and hungry/well-fed, . . . Temporal Resolution Single- vs. Spatial resolution: Multi-Population Level of Detail habitat: friendly/hostile environment Finally: What’s the Task? location of food, competitors, predators, . . . What Quantities have an Effect? what other species have to be included? how detailed do we need to model the environment?

Scientific Finally: What’s the Task? Computing I Michael Bader Discrete vs. Continuous Deterministic vs. Stochastic find a solution (find all solutions) Spatial and Temporal find the best solution (optimization problem) Resolution analyse solutions: Single- vs. Multi-Population Is it unique? How does it depend on input Level of Detail data? Finally: What’s the Task? validate the model: quantitatively vs. qualitatively correct?