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# Competing Species vs Predator-Prey Models

Final Paper Update #1

by

Tweet## Lauren Thomas

on 7 December 2012#### Transcript of Competing Species vs Predator-Prey Models

Logistic

Equations System of Equations Revised Logistic

Equations To Do List # of individuals in a population increase in a given time period values are dependent upon the species under observation Absence of predator understand variables An Analysis of Population Dynamic Models Lauren R. Thomas

Florida A&M University represents the populations closed environment Growth Rates Saturation Levels population x population y Interference Factor measure of degree to which species y interferes with species x Competing Species Predator-Prey Assumptions prey will grow at a rate proportional to current population Absence of prey predator will die out Encounters between predator & prey predator is increased prey is decreased prey predator known as Lotka-Volterra Equations developed in papers by Lotka in 1925 and by Volterra in 1926 known how to manipulate equations correctly identify which model to use realize the limitations of nature Solutions periodic solutions linearization yields a solution similar to harmonic motion Population

Equilibrium neither population is changing = equilibrium derivatives = zero Equilibria Fixed Points extinction of both species sustain current populations System Dynamics stability of fixed point at the origin (0,0) Steady State at (0,0) First Fixed Point Second Fixed Point stability of fixed point Steady State at Eigenvalues Stationary Point Jacobian Matrix the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector Eigenvalues Saddle Point Jacobian Sharks VS Fish Goal: numerically solve the P-P Model for sharks & fish Sharks = Predators

Fish = Prey Populations Fish Sharks Initial Conditions growth rate of fish in the absence of sharks death rate of sharks in the absence of their prey, fish death rate per encounter of fish with sharks efficiency of turning predated fish into sharks Parameters Fish Dependency In the absence of sharks, and the fish population is given by so that the fish population grows exponentially according to Shark Dependency In the absence of fish, and the shark population is given by so that the shark population decreases according to Nondimensionalization simplify the governing equations so that the number of undetermined constants in them is a minimum Substitute terms into original equations after non-dimensionalizing Vector Form to solve P-P Model numerically, it must be written in vector form with initial conditions In matrix-vector form this system is given by where with initial condition New Constants Problem Specifics 0.7 per year 0.5 per year 0.007 per shark per year 0.1 shark per fish a = 0.5

b = 0.6

c = 0.7143 We are given... non-dimensional constants Non-dimensionalize the total time of 50 years Solved using forward Euler method the Euler method updates the new value of y with the forward derivative to obtain Minimum fish population Maximum fish population Minimum shark population Maximum shark population 86 514 43 195 Saddle Point Saddle points are always unstable because almost all trajectories depart from them as t increases. Equilibrium Solutions correspond to no change or variation in the value of y as t increases. Critical Points the zeros of F(y) Definitions Example two species

of fish pond compete for

available food two similar species

competing for a

limited food supply closed environment Competing

Species Example foxes prey

on rabbits forest rabbits live on

vegetation two species closed environment one species (predator)

other species (prey) predator prey different food

source Predator -Prey Significance Size Birth Rate Mortality Rate Immigration Emigration (linearize) make linear or get into a linear form Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. Periodic solutions of equations are solutions that describe regularly repeating processes If a population has a constant birth rate through time and is never limited by food or disease, it has what is known as exponential growth. With exponential growth the birth rate alone controls how fast (or slow) the population grows. *Disclaimer*: this is a modification of a model completed by the Mathematics Dept. at Stanford University. In REALITY: This concludes my presentation Thanks for your attention! Questions? this fixed point is not hyperbolic

Full transcriptEquations System of Equations Revised Logistic

Equations To Do List # of individuals in a population increase in a given time period values are dependent upon the species under observation Absence of predator understand variables An Analysis of Population Dynamic Models Lauren R. Thomas

Florida A&M University represents the populations closed environment Growth Rates Saturation Levels population x population y Interference Factor measure of degree to which species y interferes with species x Competing Species Predator-Prey Assumptions prey will grow at a rate proportional to current population Absence of prey predator will die out Encounters between predator & prey predator is increased prey is decreased prey predator known as Lotka-Volterra Equations developed in papers by Lotka in 1925 and by Volterra in 1926 known how to manipulate equations correctly identify which model to use realize the limitations of nature Solutions periodic solutions linearization yields a solution similar to harmonic motion Population

Equilibrium neither population is changing = equilibrium derivatives = zero Equilibria Fixed Points extinction of both species sustain current populations System Dynamics stability of fixed point at the origin (0,0) Steady State at (0,0) First Fixed Point Second Fixed Point stability of fixed point Steady State at Eigenvalues Stationary Point Jacobian Matrix the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector Eigenvalues Saddle Point Jacobian Sharks VS Fish Goal: numerically solve the P-P Model for sharks & fish Sharks = Predators

Fish = Prey Populations Fish Sharks Initial Conditions growth rate of fish in the absence of sharks death rate of sharks in the absence of their prey, fish death rate per encounter of fish with sharks efficiency of turning predated fish into sharks Parameters Fish Dependency In the absence of sharks, and the fish population is given by so that the fish population grows exponentially according to Shark Dependency In the absence of fish, and the shark population is given by so that the shark population decreases according to Nondimensionalization simplify the governing equations so that the number of undetermined constants in them is a minimum Substitute terms into original equations after non-dimensionalizing Vector Form to solve P-P Model numerically, it must be written in vector form with initial conditions In matrix-vector form this system is given by where with initial condition New Constants Problem Specifics 0.7 per year 0.5 per year 0.007 per shark per year 0.1 shark per fish a = 0.5

b = 0.6

c = 0.7143 We are given... non-dimensional constants Non-dimensionalize the total time of 50 years Solved using forward Euler method the Euler method updates the new value of y with the forward derivative to obtain Minimum fish population Maximum fish population Minimum shark population Maximum shark population 86 514 43 195 Saddle Point Saddle points are always unstable because almost all trajectories depart from them as t increases. Equilibrium Solutions correspond to no change or variation in the value of y as t increases. Critical Points the zeros of F(y) Definitions Example two species

of fish pond compete for

available food two similar species

competing for a

limited food supply closed environment Competing

Species Example foxes prey

on rabbits forest rabbits live on

vegetation two species closed environment one species (predator)

other species (prey) predator prey different food

source Predator -Prey Significance Size Birth Rate Mortality Rate Immigration Emigration (linearize) make linear or get into a linear form Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. Periodic solutions of equations are solutions that describe regularly repeating processes If a population has a constant birth rate through time and is never limited by food or disease, it has what is known as exponential growth. With exponential growth the birth rate alone controls how fast (or slow) the population grows. *Disclaimer*: this is a modification of a model completed by the Mathematics Dept. at Stanford University. In REALITY: This concludes my presentation Thanks for your attention! Questions? this fixed point is not hyperbolic