Rickers Population Model The Study of the Existence and Stability of - PowerPoint PPT Presentation

Motivation Preliminaries Examples and Experimentations Conclusion Ricker’s Population Model The Study of the Existence and Stability of Equilibria Within an Ecosystem Willie Bell James Boffenmyer Scott Dean Andrew Stewart LSU Summer Mathematics Integrated Learning Experience, 2010 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries Examples and Experimentations Conclusion Outline Motivation 1 History Applications Preliminaries 2 Necessary Definitions Necessary Theorems Examples and Experimentations 3 2 Arbitrary Cases Conclusion 4 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries History Examples and Experimentations Applications Conclusion Outline Motivation 1 History Applications Preliminaries 2 Necessary Definitions Necessary Theorems Examples and Experimentations 3 2 Arbitrary Cases Conclusion 4 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries History Examples and Experimentations Applications Conclusion The Originator Dr. Bill Ricker (1908-2001) x n + 1 = x n 훿 e r − x n − k (1) r , 훿, x 0 ∈ ℝ + , n ∈ ℤ + Best known for the Ricker model, which he developed in his studies of stock and recruitment in fisheries. Throughout his lifetime, Dr. Ricker published 296 papers and books, 238 translations, and 148 scientific and literary manuscripts.” Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries History Examples and Experimentations Applications Conclusion Outline Motivation 1 History Applications Preliminaries 2 Necessary Definitions Necessary Theorems Examples and Experimentations 3 2 Arbitrary Cases Conclusion 4 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries History Examples and Experimentations Applications Conclusion The Applications Fishery Sciences Biological Sciences Human Population Modeling Any field of science that involves a population of species, Ricker’s model can be applied. However, these are only a few of the areas of which Ricker’s can be used. Day after day, more applications are being developed for this model. Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries History Examples and Experimentations Applications Conclusion The Applications Fishery Sciences Biological Sciences Human Population Modeling Any field of science that involves a population of species, Ricker’s model can be applied. However, these are only a few of the areas of which Ricker’s can be used. Day after day, more applications are being developed for this model. Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries History Examples and Experimentations Applications Conclusion The Applications Fishery Sciences Biological Sciences Human Population Modeling Any field of science that involves a population of species, Ricker’s model can be applied. However, these are only a few of the areas of which Ricker’s can be used. Day after day, more applications are being developed for this model. Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries History Examples and Experimentations Applications Conclusion The Applications Fishery Sciences Biological Sciences Human Population Modeling Any field of science that involves a population of species, Ricker’s model can be applied. However, these are only a few of the areas of which Ricker’s can be used. Day after day, more applications are being developed for this model. Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries Necessary Definitions Examples and Experimentations Necessary Theorems Conclusion Outline Motivation 1 History Applications Preliminaries 2 Necessary Definitions Necessary Theorems Examples and Experimentations 3 2 Arbitrary Cases Conclusion 4 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries Necessary Definitions Examples and Experimentations Necessary Theorems Conclusion Definitions Definition ¯ x is an equilibrium of the equation x n + 1 = f ( x n ) , n = 0 , 1 , ... (2) if x = f (¯ ¯ x ) . The correpsonding solution { ¯ x n } such that x n = ¯ ¯ x is called also a constant solution or steady-state solution . Also, in such cases we say that ¯ x is a fixed point of the function f . Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries Necessary Definitions Examples and Experimentations Necessary Theorems Conclusion Definitions Definition (Stability) (i) The equilibrium point ¯ x of Eq. (2) is called (locally) stable if for every 휖 > 0 there exists 훿 > 0 such that ∣ x 0 − ¯ ∣ x n − ¯ x ∣ < 훿 implies x ∣ < 휖 for n ≥ 0 . Otherwise, the equilibrium ¯ x is called unstable. (ii) The equilibrium point ¯ x of Eq. (2) is called (locally) asymptotically stable ( LAS) if it is stable and there exists 훾 such that ∣ x 0 − ¯ n →∞ ∣ x n − ¯ x ∣ < 훾 x ∣ = 0 . implies lim Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries Necessary Definitions Examples and Experimentations Necessary Theorems Conclusion Outline Motivation 1 History Applications Preliminaries 2 Necessary Definitions Necessary Theorems Examples and Experimentations 3 2 Arbitrary Cases Conclusion 4 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries Necessary Definitions Examples and Experimentations Necessary Theorems Conclusion Theorems Theorem Let ¯ x be an equilibrium of the difference equation (1) where f is a continuously differentiable function at ¯ x . (i) If � < 1 � � � f ′ (¯ x ) then the equilibrium ¯ x is locally asymptotically stable . (ii) If � > 1 � � � f ′ (¯ x ) then the equilibrium ¯ x is unstable . Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion Outline Motivation 1 History Applications Preliminaries 2 Necessary Definitions Necessary Theorems Examples and Experimentations 3 2 Arbitrary Cases Conclusion 4 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion 2 Arbitrary Cases For the two particular cases we will present, we will briefly describe what will happen when: 0 < 훿 ≤ 1 훿 > 1 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion 2 Arbitrary Cases For the two particular cases we will present, we will briefly describe what will happen when: 0 < 훿 ≤ 1 훿 > 1 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion 2 Arbitrary Cases For the two particular cases we will present, we will briefly describe what will happen when: 0 < 훿 ≤ 1 훿 > 1 Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion Set Up In order to better understand the difference equation x n + 1 = x n 훿 e r − x n , we represent the model with the function f ( x ) = x 훿 e r − x . We can use this function to better solve for equilibria and analyze stability. Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion Finding Equilibria In order to solve for equilibria of our equation, we must set the function f ( x ) = x . So x 훿 e r − x = x x 훿 e r − x − x = 0 x ( x 훿 − 1 e r − x − 1 ) = 0 So an equilibrium exists at x = 0 and for the solution(s) of the equation x 훿 − 1 e r − x − 1 = 0. Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion Establishing ¯ x x 훿 − 1 e r − x − 1 = 0 ⇒ 0 = x 1 − 훿 − e r − x = g ( x ) We know that x ∈ [ 0 , ∞ ) . g ( 0 ) = − e r x →∞ g ( x ) = ∞ lim ⇔ g ( x ) = 0 has at least one solution on ( 0 , ∞ ) by the IVT . Willie B James B Scott D Andrew S Ricker’s Population Model

Motivation Preliminaries 2 Arbitrary Cases Examples and Experimentations Conclusion Establishing ¯ x We can now use g ′ ( x ) to analyze the behavior of g ( x ) on [ 0 , ∞ ) . g ′ ( x ) = ( 1 − 훿 ) x − 훿 + e r − x Since 훿 > 0 , r > 0 , g ′ ( x ) > 0 , ∀ x ∈ ( 0 , ∞ ) , ⇔ g ( x ) ↑ on ( 0 , ∞ ) , ⇔ g ( x ) has exactly one solution ¯ x . Willie B James B Scott D Andrew S Ricker’s Population Model