Michel 1 Nonlinear Systems with Applications in Ecology By: Jerod Michel

Chapter 0 Introduction

The study of mathematical models in ecology, which is the study of the relationship between species and their environment, is rapidly expanding. These models help us understand the dynamic relationship between predator and prey, competing species, and mutual species. These models also help us understand other ecological areas such as renewable resource management, evolution of pesticide resistant strains, control of pests, multi-species societies, plant-herbivore systems, etc. (Murray 63) The following is an expository paper targeting junior, undergraduate math students, about continuous models for interacting populations. It is recommended that readers have had courses in calculus, linear algebra and differential equations. Following this introduction will be a discussion of population growth, and then linear and nonlinear systems, and how they relate to ecology. When discussing nonlinear systems in the phase plane, most of the ideas will relate to a concept called linearization. Finally, a transition will be made into discussing the application of these techniques to situations in nature where a predator-prey situation is present. The application to be explored is the Lotka-Volterra Equations (chapter 2). Population Growth Attempts to understand population models date back to the middle ages. It was often the case that human populations were the focus. In 1798 Thomas Malthus came up with the first-order, linear population growth model commonly referred to as the Malthusian Growth Model (Nagle 51). This model for a population is,

dp = kp , p( 0 ) = p0 , dt

(0.1)

where k = a - b for a, b R , and a and b are effectively birth and death rates. This equation is separable and has the solution,

p( t ) = p0e kt .

(0.2)

This model has serious drawbacks, e.g. it assumes that the only factors in predicting the growth of a population are the past and future growth estimates. In 1836, Verhulst proposed a model that limits itself when it becomes too large:

dp = - Ap( p - p1 ) , p( 0 ) = p0 , dt

(0.3)

where âˆ’ A( p âˆ’ p1 ) is the per capita birth rate (which is dependent on p), and p1 is the carrying capacity of the environment, which is determined by the available sustaining resources. This nonlinear model, commonly referred to as the Logistic Growth Model, is often more realistic than the Malthusian Growth Model because it considers a state of overpopulation (Nagle 55). Again separating variables, (0.3) yields the solution:

Michel 2

p( t ) =

p0 p1 . p0 + ( p1 − p0 )e −Ap1t

(0.4)

The long-time behavior of this solution (i.e. letting t → ∞ ) tends to p1 , the so-called carrying capacity for the species. Before considering models of two or more species we need to consider some background mathematics. Most models for such systems are nonlinear, but linear systems come into play quite often when studying nonlinear systems. Since the application discussed in the later part of this paper involves a nonlinear system, it is in order to first discuss linear systems. Linear Systems

dx = f ( x ) , where x ∈ R n is called a linear system of dt dimension n, if f is a linear mapping (Chapman 76). Definition 0.5 A system

If the mapping f : R n → R n , where R n = {( x1 ,...,x n ) x i ∈ R , i = 1, 2 ,...} is linear, then it can be written, ⎡ f1 ( x1 ,..., x n ) ⎤ ⎡a11 ⎢ ⎥ ⎢ ⋅ ⎢ ⎥ ⎢ ⎥=⎢ f( x ) = ⎢ ⋅ ⎢ ⎥ ⎢ ⋅ ⎢ ⎥ ⎢ ⎢⎣fn ( x1 ,..., x n )⎥⎦ ⎢⎣ an1 Then

⋅ ⋅ ⋅ a1n ⎤ ⎡ x1 ⎤ ⎥ ⎢ ⋅ ⎥ ⋅ ⎥ ⎢ ⎥ ⎥⋅⎢ ⋅ ⎥. ⋅ ⎥ ⎢ ⎥ ⋅ ⎥ ⎢ ⋅ ⎥ ⋅ ⋅ ⋅ ann ⎥⎦ ⎢⎣ x n ⎥⎦

(0.6)

dx becomes, dt dx = A⋅x , dt

(0.7)

where A is the coefficient matrix (Chapman 78). e.g. Consider the system of equations,

dx 1 dx 2 = x 2 and = x1 , dt dt

(0.8)

which can be transformed into the decoupled equations,

dy 1 dy 2 = y 1 and = −y 2 dt dt

(0.9)

x1 = y 1 + y 2 and x 2 = y 1 − y 2 .

(0.10)

when we write,

Then, in matrix form, we get,

Michel 3

⎡ dxdt1 ⎤ ⎡0 1⎤ ⎡ x1 ⎤ ⎢ dx2 ⎥ = ⎢ ⎥ ⋅ ⎢x ⎥ , 1 0 ⎣ ⎦ ⎣ 2⎦ ⎣ dt ⎦

(0.11)

⎡ x1 ⎤ ⎡1 1 ⎤ ⎡ y 1 ⎤ ⎢ x ⎥ = ⎢1 − 1⎥ ⋅ ⎢ y ⎥ , and ⎣ 2⎦ ⎣ ⎦ ⎣ 2⎦

(0.12)

⎡ dydt1 ⎤ ⎡1 0 ⎤ ⎡ y 1 ⎤ ⎢ dy 2 ⎥ = ⎢ ⎥ ⋅ ⎢y ⎥ . 0 − 1 ⎣ ⎦ ⎣ 2⎦ ⎣ dt ⎦

(0.13)

Therefore, ⎡0 1 ⎤ ⎡1 1 ⎤ ⎡1 0 ⎤ A=⎢ , M= ⎢ and B = ⎢ ⎥ ⎥ ⎥. ⎣1 0 ⎦ ⎣1 − 1⎦ ⎣0 − 1⎦

(0.14)

Observe that A ⋅ M = M ⋅ B , and call B = M −1 ⋅ A ⋅ M the similar matrix to A . Proposition 0.15 Let A be a 2 × 2 matrix. Then there exists an invertible matrix M such that B = M−1 ⋅ A ⋅ M is one of the types: ⎡ λ1 0 ⎤ (1) ⎢ ⎥ , λ1 > λ2 ; ⎣ 0 λ2 ⎦ 1⎤ ⎡λ (3) ⎢ 0 ⎥, ⎣ 0 λ0 ⎦

⎡ λ0 0 ⎤ (2) ⎢ ⎥ ⎣ 0 λ0 ⎦ ⎡α − β ⎤ (4) ⎢ ⎥, β > 0 ⎣β α ⎦

(0.16)

where λ0 , λ1 , λ2 ,α , β ∈ R (Chapman 81). The matrix B is the Jordan form of A, and the eigenvalues of A (and B) satisfy:

γ A ( λ ) = λ2 − tr ( A )λ + det( A ) = 0 .

(0.17)

Here tr( A) = a11 + a 22 is the trace of A and det(A) = a11a 22 − a12 a21 is the determinant of A. It can be shown that the eigenvalues are,

λ1 = 12 ( tr ( A ) + Δ ) and λ2 = 12 ( tr ( A ) − Δ ) where Δ = ( tr ( A )) 2 −4 det( A ) is the discriminant of the general solution to (0.17). If Δ > 0 then the eigenvalues are real and distinct, so, ⎡ λ0 B=⎢ ⎣0

0⎤ , λ1 > λ2 ; is the Jordan matrix (0.16.1). λ0 ⎥⎦

If Δ = 0 , the eigenvalues are equal, and either A is diagonal and B = A, and is the Jordan matrix (0.16.2); or A is not diagonal and, ⎡ λ0 B=⎢ ⎣0

1⎤ is the Jordan matrix (0.16.3). λ0 ⎥⎦

(0.18)

Michel 4

If Δ < 0 , then the eigenvalues are complex and, ⎡α A= ⎢ ⎣β

− β⎤ , β > 0 ; is the Jordan matrix (0.16.4). α ⎥⎦

Since the audience targeted by this paper, junior and senior undergraduate math students, is assumed to have had courses in linear algebra and differential equations, this will mark the extent of the linear system content of the paper. From this point on, the discussion will include content from nonlinear systems and ecological applications. Chapter 1 Nonlinear Systems

Since fixed points (or steady states) 1 are much easier to analyze in a linear system, most concepts and definitions from this section will be centered around a concept called linearization. This concept abridges the process of analyzing nonlinear systems by recognizing how closely trajectories around the fixed points of nonlinear systems resemble those of linear systems. Biologists and ecologists often rely on this technique when studying two or more interacting populations because the mathematical models they rely on involve nonlinear systems which are difficult or impossible to solve directly. Before discussing linearization, it is in order to first discuss some basic definitions and ideas about nonlinear systems. dx In this chapter we consider the systems = X( x ) , x ∈ S ⊆ R 2 , where X is a dt continuously differentiable, nonlinear function. Definition 1.1 A neighborhood N of a point x 0 ∈ R 2 is a subset of R 2 containing a disc

{x

x-x 0

< r } for some r > 0 (Chapman 97).

Definition 1.2 The part of the phase portrait of a system that occurs in a neighborhood N of x 0 is called the restriction of the phase portrait to N (Chapman 99). Restrictions are important in nonlinear systems because a restriction of the complete phase portrait to a neighborhood of x 0 may be any size we please. Call such a restriction the local phase portrait at x 0 . Consider a restriction of a simple, linear system to a neighborhood N of the origin. There must exist a neighborhood N 0 ⊆ N such that the restriction to the phase portrait to N 0 is qualitatively equivalent to the complete phase portrait of the simple linear system because there is a continuous bijection between N 0 and R 2 which maps the restriction N 0 onto the complete phase portrait (Chapman 99). For example the phase portrait for,

1

Fixed points, or steady states, are solutions of the form x( t ) ≡ c = ( c 1 ,c 2 ) , which arise when

X 1 ( c 1 ,c 2 ) = 0 and X 2 ( c1 ,c 2 ) = 0 .

Michel 5

dx 1 dx 2 = 3x1 + 4 x 2 and = âˆ’3x1 âˆ’ 3 x 2 is pictured below. dt dt

(1.3)

(1.3.1) Here is the restriction of (1.3.1) to N = {( x1 , x 2 ) a < x1 < b , c < x 2 < d ; a ,d < 0 and b ,c > 0 } ,

(1.3.2) where the dashed line is the critical trajectory T (to be explained below). Finally, the restriction to the neighborhood N 0 = { x x is inside T }, which is qualitatively equivalent to (1.3.1), is here.

(1.3.3)

Michel 6 Notice that (1.3.3) correlates qualitatively to (1.3.1). However, since nonlinear systems may have more than one fixed point (especially when modeling biological systems with more than one steady state), local phase portraits cannot always determine the complete phase portrait. The trajectories of (1.3) are called limit cycles , which will be discussed in more depth shortly. Linearization at a Fixed Point Definition 1.4 Suppose

h1 =

dy

= Y( y ) can be written such that

dt

dy 1 dy 2 = ay 1 + by 2 + g 1 ( y 1 , y 2 ) , and h2 = = cy 1 + dy 2 + g 2 ( y 1 , y 2 ) dt dt

(1.4.1)

where g i is a nonlinear function of y 1 and y 2 , i = 1, 2. Then, using a two-variable Taylor series expansion, we get,

⎡ h1 ( y 1 , y 2 ) = h1 ( 0,0 ) + ⎢ y 1 ⎢⎣ ⎡ + ⎢ y 12 ⎢ ⎣

∂ 2 h1 ∂y 2 ∂y 1

⎡ + ⎢ y 13 ⎢ ⎣

∂ 3h1

( 0 ,0 )

2!

∂ y13

( 0 ,0 ) 3!

≈ y1

∂h1 ∂y1

( 0,0 ) 1!

+ y1 y 2

∂ 2 h1 ∂y1∂ y 2

( 0 ,0 )

2!

∂ 3h1

+ y 12 y 2

+ y2

∂ y 23

( 0 ,0 ) 3!

∂h1 ∂y 2

( 0,0 ) ⎤ ⎥ 1! ⎥⎦ ∂ 2 h1

+ y 2 y1

∂ y12

2!

∂ 3h1

+ y 22 y 1

∂ y1∂ y 22

∂h1 ( 0 ,0 ) ∂h ( 0,0 ) + y2 1 ∂y 1 ∂y 2

= ay 1 + by 2 is the linearization of

( 0,0 )

3!

∂ 2 h1

( 0 ,0 ) +y

∂ 3h1

+ y 23

( 0 ,0 ) ⎤ ⎥ ⎥ 2! ⎦

2 2 ∂y 2 2

( 0,0 ) ⎤ ⎥ + h .o .t . ⎥ 3! ⎦

∂ y12∂ y 2

(1.4.2)

dy 1 at the origin where h.o.t. abbreviates dt

higher order terms . (Britton 268). Similarly, h2 ( 0,0 ) ≈ y 1

∂h2 ( 0,0 ) ∂h ( 0,0 ) + y2 2 ∂y 1 ∂y 2

= cy 1 + dy 2 . Notice that g i approaches zero because of the behavior of higher order derivatives of nonlinear terms when using the Taylor series expansion, i.e. if y 1 and y 2 are small, then products of them are very small, and the nonlinear part of the expansion may be safely neglected.

Michel 7 The above definition can also be applied to fixed points not at the origin by introducing local coordinates.

dx = X( x ) where x = ( x1 , x 2 ) . dt Let y 1 = x1 − α and y 2 = x 2 − β be a set of Cartesian coordinates for the phase plane with their origin at ( x1 , x 2 ) = ( α , β ) . These are called local coordinates at ( α , β ) . Thus, Suppose ( α , β ) is a fixed point of a nonlinear system

dy i dx i = = X i ( y 1 + α , y 2 + β ) , i = 1, 2 dt dt

(1.4.3)

where X 1 and X 2 are component functions of x . If we define,

Yi ( y 1 , y 2 ) = X ( y 1 + α , y 2 + β ) , then

dy i becomes, dt

(1.4.4)

⎡ ∂X i ⎤ ∂X i X i ( y1 + α ,y 2 + β ) = X i ( α , β ) + ⎢ y1 ( α ,β ) + y2 ( α , β )⎥ ∂x 2 ⎣ ∂x1 ⎦ ⎡y 2 ∂2 Xi ⎤ y1 y 2 ∂ 2 X i y 2 y1 ∂ 2 X i y 22 ∂ 2 X i +⎢ 1 ( α , β ) + ( α , β ) + ( α , β ) + ( α , β )⎥ 2 2 2! ∂x1 ∂x 2 2! ∂x 2 ∂x1 2! ∂x 2 ⎣ 2! ∂x1 ⎦ ⎡ y 13 ∂ 3 X i ⎤ y 12 y 2 ∂ 3 X i y 22 y 1 ∂ 3 X i y 23 ∂ 3 X i +⎢ ( α , β ) + ( α , β ) + ( α , β ) + ( α , β ) ⎥ + h .o .t 3 3! ∂x12 ∂x 2 3! ∂x 22 ∂x1 3! ∂x 23 ⎣ 3! ∂x1 ⎦ ≈ y1

∂X i ∂X i ( α ,β ) + y 2 ( α ,β ) ∂x1 ∂x 2

(1.4.5)

where, again, the nonlinear part of the expansion may be safely discarded (because products of small terms are very small). This is called the linearization at ( α , β ) and is further simplified by defining,

a=

∂X 1 ∂X 1 ∂X 2 ∂X 2 , b= , c= , d= , ∂x1 ∂x 2 ∂x1 ∂x 2

(1.4.6)

all of which are evaluated at ( α , β ) (Britton 268). Thus, we get the matrix form of dy = Ay where, dt ⎡a b ⎤ . A=⎢ ⎥ ⎣c d ⎦ ( x1 ,x 2 )=( α ,β )

(1.4.7)

Michel 8

Theorem 1.5 Let the nonlinear system

dy = Y( y ) have a fixed point at y = 0. Then in a dt

neighborhood of the origin the phase portraits of the system and its linearization are qualitatively equivalent provided the linearized system is not a center 2 (Chapman 102). Fixed points, when present in a nonlinear system, represent relationships between two or more functions. In a continuous model for two or more interacting populations, the fixed points are referred to as steady states, and they represent growth equilibrium in the applications discussed in this paper, i.e. the state at which two populations can coexist without changing in size with respect to time. However, there are different types of stability, and these are addressed below (Murray 64). Definition 1.6 If there is a neighborhood around a fixed point of the system

dx = X( x ) dt

such that every trajectory passing through it tends to it as t tends to infinity, then the fixed point is asymptotically stable. dx Definition 1.7 A fixed point of the system = X( x ) which is stable but not dt asymptotically stable is said to be neutrally stable (Chapman 105). There are many examples of neutrally stable fixed points. Consider the nontrivial fixed point of the equations:

dx 1 dx 2 = x1 ( a − bx 2 ) and = − x 2 ( c − dx 1 ) with, dt dt

(1.6.1)

a, b, c, d > 0. Nuetral stability of the fixed point at ( dc , ab ) follows from the existence of neighborhoods N and N 0 which satisfy the definition of stability. Clearly, ( dc , ab ) is a fixed point:

dx 1 c a = ( )( a − b( )) dt d b =(

c )( a − a ) d

=(

c )( 0 ) d

= 0.

dx 2 a c = −( )( c − d ( )) dt b d = −(

a )( c − c ) b

= −(

a )( 0 ) b

= 0.

The system (1.6.1) is one of the earliest and simplest predator-prey models. Prey is dx 2 dx 1 represented by and predator by . The equations are called the Lotka-Volterra dt dt Equations and will be discussed in the next chapter in more depth.

2

A center is a system whose phase portrait consists of a continuum of concentric circles.

Michel 9

dx = X( x ) which is neither neutrally stable dt nor asymptotically stable is said to be unstable. Definition 1.7 A fixed point of the system

Practically this means that any deflection from the fixed point will likely lead to an evolution of the trajectory away from the fixed point. dx Definition 1.8 Any point in the phase plane of the system = X( x ) which is not a dt fixed point is said to be an ordinary point. Thus, if x 0 is an ordinary point, then X( x 0 ) ≠ 0 , and by the continuity of X, there exists a neighborhood of x 0 containing only ordinary points. In the phase plane, there are several interesting types of steady states. When two or more interacting populations have periodic solutions that correspond to closed curves in the phase plane, an interesting steady state for the interacting populations arises. These are called limit cycles, but before discussing them we will need the preliminary definitions below. Definition 1.9 A continuously differentiable function f : D( ⊆ R 2 ) → R is called a first dx integral of the system , x ∈ S ⊆ R 2 on the region D ⊆ S if f ( x( t )) is constant for any dt solution x( t ) of the system (Chapman 107). Proposition 1.10 Let ( x1 , x 2 ) = ( α , β ) be a steady state of the first integral f ( x( t )) . Then there is a neighborhood of ( α , β ) in which the level curves of f ( x( t )) are closed (Chapman 107). Definition 1.11 A system that has a first integral on the whole of the plane is called conservative. Limit Cycles Definition 1.12 A closed trajectory C in a phase portrait is called a limit cycle if it is isolated from all other closed trajectories, i.e. if there is a tubular neighborhood of C which contains no other closed trajectories. We can portray this definition by comparing the limit cycle with the center. It is easy to come up with examples using polar coordinates. e.g. Show the system, 1 dx 1 = − x 2 + x1 [ 1 − ( x12 + x 22 ) 2 ] (= Þ) , and dt

(1.13)

dx 2 = x1 + x 2 [ 1 − ( x12 + x 22 ) ] (= ß) dt 1 2

has a limit cycle given by x12 + x 22 = 1 . In polar coordinates x1 = r cos θ and x 2 = r sin θ . If r 2 = x12 + x 22 then we get

(1.13.1)

Michel 10

dr dθ = r ( 1 − r ) and = 1 (see appendix B). dt dt

(1.13.4)

Consider r ( t ) ≡ 1 and θ( t ) = t . Then θ( t ) = t ⇒ H = 1 (see appendix B), and r ( t ) ≡ 1 and θ ( t ) = t gives a closed trajectory consisting of the circle x12 + x 22 = 1 traversed in a dθ dr counterclockwise direction at constant angular speed = 1 . For 0 < r < 1 , > 0 and dt dt dr trajectories must spiral outwards toward r = 1. However, r > 1 yields < 0 and dt trajectories must spiral inwards as t increases.

(1.13.5) Definition 1.14 Limit cycles do not all behave in the same way – there are three different types: 1) the stable limit cycle, where trajectories spiral into the closed orbit from either side as t → ∞ such as in (1.13), 2) the unstable limit cycle where trajectories spiral away from the closed orbit on both sides as t → ∞ , and 3) the semi-stable limit cycle where trajectories spiral toward the closed trajectory on one side and spiral away from it on the other (Chapman 120). This brings an end to the strictly theoretical content of this paper. From this point on, the paper will be focused on applying the above concepts and techniques to ecological phenomena. It is conventional to use slightly different notation when describing ecological situations with differential equations. To help readers adjust, the necessary references will be in place. Chapter 2 The Lotka-Volterra Equations

Before introducing an application, an overview of how the application will be presented is in order. Since mathematical modelers must make assumptions about nature to simplify models and make them more user-friendly for scientists with less of a mathematical background, it is often helpful to first list the assumptions and then discuss the application. Following the assumptions will be a brief description of the derivation of the application, trivial and nontrivial steady states, linearization of the system at its nontrivial steady state, an assessment of the stability of its steady state (which, in ecological terms, addresses how quickly population growth will leave or come

Michel 11 back to equilibrium when - not at but - close to equilibrium). A worked example of the model applied to a situation in nature would normally follow, however, space does not allow for it. Shall we begin? The Lotka-Volterra Equations Assume: 1) (implicit) Predator-prey interactions are the only determinants of population dynamics, 2) The prey is limited by the predator only, and in its absence would exponentially, 3) The functional response3 of the predator is linear (in U), 4) There is no interference4 between predators in finding prey, 5) In the absence of the prey, the predator dies off exponentially, 6) Every prey death contributes directly to the growth of the predator population (Britton 55). Let a , b , c , d ∈ R + , let U be the prey and V the predator. Also let p and q be catchability coefficients and E be the effort of an outside force preying on both populations U and V. Then the Lotka-Volterra equations are:

dU dV = aU − pEU − cUV and = ecUV − qEV − bV . dτ dτ The term steady-state refers to when and (

(2.1)

dU dV = = 0. Then (0, 0) is a trivial steady-state dτ dτ

qE + b a − pE , ) is a non-trivial steady-state (see appendix C for proof). ec c

For E >

a qE + b a − pE , the outside force increases U * = (prey) and decreases V * = p ec c

(predator). We will now nondimensionalize5 this system. Let u = let v =

U U ecU and = qE + b = * U qE + b ec

V V cV . Then t = ( a − pE )τ is the new time variable. So now we get: = a − pE = * V a − pE c dU dτ = [( a − pE )U − cUV ]( ) dt dt = U( 1 − V * ) ,

and we also get

dV dτ = [ ecUV − ( qE + b )V ]( ) dt dt

= αV ( U * − 1 ) . Thus, when the system is nondimensionalized we get:

3

The term of predation is linear in U. Encounters between predators reduce efficiency in searching for prey. 5 Nondimensionalization refers to reducing the number of parameters to dimensionless groupings which determine the dynamics (Murray 5). 4

Michel 12

du dv = u( 1 − v ) and = α ⋅ v( u − 1 ) (see appendix D). dt dt

(2.2)

dv du dv α ⋅ v ( u − 1 ) by we get = which is in the (u, v)-plane du u( 1 − v ) dt dt (also known as the phase plane). This system has a first integral by (1.9) (see appendix A) thereby producing a closed trajectory by (1.10). Thus when we divide

Call hu =

du dv and hv = . Then, by (1.4) we get: dt dt

∂h ⎡ ∂h ⎤ hu ( u ,v ) = hu ( 0,0 ) + ⎢u u ( 0,0 ) + v u ( 0,0 )⎥ + h .o .t . ∂v ⎣ ∂u ⎦ ≈ ( 1 )u + ( 0 )v , and ∂h ⎡ ∂h ⎤ hv ( u ,v ) = hv ( 0 ,0 ) + ⎢u v ( 0,0 ) + v v ( 0,0 )⎥ + h.o .t . ∂v ⎣ ∂u ⎦

≈ ( 0 )u + ( −α )v , which is further simplified in the matrix form dh = Ah where, dt ⎛1 0 ⎞ A = ⎜⎜ ⎟⎟ , and h = (u, v). ⎝0 − α⎠

(2.3)

Since one eigenvalue is greater than zero and the other is less, (0, 0) is a saddle point that is linearly unstable by (1.14). In ecological terms this means that small deflections from the steady state result in rapid growth of the populations away from the steady state. This makes sense since the steady state (0, 0) represents no prey or predators being present. Linearizing about the steady state (1, 1) by setting u = 1 + x and v = 1 + y , by (1.4) we get: ∂h ⎡ ∂h ⎤ hu ( x + 1, y + 1 ) = hu ( 1,1 ) + ⎢ x u ( 1,1 ) + y u ( 1,1 )⎥ + h.o .t . ∂v ⎣ ∂u ⎦ ≈ ( 0 )x + ( −1 )y , and ∂h ⎡ ∂h ⎤ hv ( x + 1, y + 1 ) = hv ( x , y ) + ⎢ x v ( 1,1 ) + v v ( 1,1 )⎥ + h.o .t . ∂v ⎣ ∂u ⎦ ≈ ( α )x + ( 0 )y , which, in the matrix form ⎛1 0 ⎞ A = ⎜⎜ ⎟⎟ . ⎝0 − α⎠

dh = Ah , yields: dt (2.4)

Michel 13 These eigenvalues are purely imaginary so the equations linearized about the steady state have periodic solutions and the steady state is a center. But depending on the nonlinear terms, trajectories could spiral into or away from the steady state and by (1.7) the steady state is neutrally stable. In ecological terms this suggests that small perturbations will move solutions onto trajectories that either bring populations back to the steady state or bring them away depending on the nonlinear terms of the system. This occurs whenever systems have a first integral (see definition 1.9) because there is a closed trajectory in the phase plane. Chapter 3 Conclusion

The benefits of understanding nonlinear systems become clear in ecology when discussing relationships between two growing populations. By linearizing a system at its steady states, one can predict whether a population will increase (decrease) in size when - not on but - close to its steady states. Since predator-prey interactions are oscillatory, it is mathematically beneficial to assume situations involving two interacting populations are periodic. However, in realistic predator-prey situations this may be too great of an assumption because the solutions relating the true growth of each population may be chaotic and not periodic. Simple predator-prey models such as (2.1) often assume that the prey is limited only by the predator. Realistically there must always be some self-limitation present, but if the predator limits the population enough, the self-limitation phenomenon may be safely neglected. Another phenomenon often neglected in models is the interference between predators even though common in the trophic web6 (Murray 315). When modifying a simple predator-prey model, mathematicians often try replacing a linear component of the system with a nonlinear component. The LotkaVolterra model was modified by Dunbar in 1983 by replacing the Malthusian growth function for prey with a logistic growth function. He also modified the application to consider spatial variations in the populations as the predators move to catch the prey (Murray 315). Another early, simple application used to describe a similar situation to predatorprey is the Host-parasitoid Equations. Parasitoids are creatures that have at least one free-living and one parasitic stage in their life cycle. The adult lays eggs in the larval stage of some host such as a caterpillar or butterfly (Britton 81). This model is not represented by differential equations but difference equations7; however, steady states that are qualitatively equivalent to those of predator-prey models occur and the process of linearization can be used to analyze the steady states just the same. Because there is limitless room for improvement and modifications, mathematical ecology remains a rapidly expanding field.

6

The trophic web refers to the web of all interacting species (Murray 63). A first-order difference (or recurrence) equation N t +1 = f ( N t ) is solved with initial condition (Britton 257). 7

N 0 given

Michel 14 Appendix A. So

dx1 dt dx 2 dt

=

x1( a − bx 2 ) dx x ( a − bx 2 ) ⇒ 1 = 1 − x 2 ( c − dx 1 ) dx 2 − x 2 ( c − dx 1 ) ⇒

∫

⇒∫

c − dx 1 a − bx 2 dx 1 = −∫ dx 2 x1 x2

c a dx1 − ∫ ddx1 = ∫ dx 2 + ∫ bdx 2 x1 x2

⇒ ( c(ln x1 + c 0 ) − dx 1 + d 0 ) = ( −a(ln x 2 + a0 ) + bx 2 + b0 ) ⇒ c ln x1 + a ln x 2 − dx 1 + bx 2 = H . B. In polar coordinates x1 = r cos θ and x 2 = r sin θ . If r 2 = x12 + x 22 then we get 2r

dx dx 2 dr , and then we substitute Þ and ß to get = 2 x1 1 + 2 x 2 dt dt dt

2r

dr = 2 x1 [ − x 2 + x1 ( 1 − r )] + 2 x 2 [ x1 + x 2 ( 1 − r )] dt

⇒

dr 1 = [ −2 x1 x 2 + 2 x12 ( 1 − r ) + 2 x1 x 2 + 2 x 22 ( 1 − r )]( r −1 ) dt 2 = [ x12 ( 1 − r ) + x 22 ( 1 − r )] r −1 = ( x12 + x 22 )( 1 − r )r −1 = r ( 1 − r ) (=

x dr dθ ), and for , we get from tan θ = 2 that x1 dt dt

sec 2 θ

dx dx dθ = x1− 2 ( 2 x1 − 1 x 2 ) and then we substitute ß and Þ to get dt dt dt

sec 2 θ

dθ = x1− 2 [( x1 + x 2 ( 1 − r ))x1 − ( − x 2 + x1 ( 1 − r ))x 2 ] dt = x1−2 [( x12 + x 2 x1 ( 1 − r )) − ( − x 22 + x1 x 2 ( 1 − r ))] = x1−2 ( x12 + x 22 )

⇒

1 dθ = x1− 2 r 2 cos 2 θ dt

Michel 15

dθ x12 ⇒ = 2 dt x1 = 1. Thus

dr dθ = r ( 1 − r ) and = 1. dt dt

Consider r ( t ) ≡ 1 and θ ( t ) = t (Notice

∫ dθ = ∫ dt ⇒ ln t + C

0

= ln θ + C1

⇒ C 0 t = C1 θ ⇒ θ ( t ) = Ht . C. WTS (

qE + b a − pE dU dV and . , ) is a steady-state for ec c dτ dτ

Consider

dU qE + b qE + b qE + b = a( ) − pE ( ) − cV ( ) dτ ec ec ec = =

Also,

( aqE + ab ) − ( pEqE + pEb ) − ( cVqE + cVb ) ec aqE + ab − pEqE − pEb − cVqE − cVb . ec

dV a − pE a − pE a − pE = ecU ( ) − qE ( ) − b( ) dτ c c c =

ecUa − ecUpE − qEa + qEpE − ba + pEb . c

dU aqE + ab − pEqE − pEb − c( Now we get = dτ ec =

a − pE c

)qE − c( a−cpE )b

aqE + ab − pEqE − pEb − aqE + pEqE − ab + pEb ec

= 0. Also,

dV ecUa − ecUpE − qEa + qEpE − ba + pEb = dτ c ec ( qEec+ b )a − ec ( qEec+ b )pE − qEa + qEpE − ba + pEb = c

Michel 16

qEa + ba − qEpE − bpE − qEa + qEpE − ba + pEb c

=

= 0. Therefore (

qE + b a − pE dU dV and . Q.E.D. , ) is a steady-state for ec c dτ dτ

D. WTS

dU dV dU dτ = U ( 1 − V * ) and = αV [ U * − 1 ] . So = [( a − pE )U − cUV ]( ) dt dt dt dt = [( a − pE )U − cUV ][

1 ] a − pE

(since if t = ( a − pE )τ then =U −[

c ]UV a − pE

= U( 1 −

cV ) a − pE

dτ 1 = ) dt a − pE

= U( 1 − V * ) . Also, we get

dV dτ = [ ecUV − ( qE + b )V ][ ] dt dt = [ ecUV − ( qE + b )V ][ =

1 ] by (*) a − pE

ecUV qE + b − V a − pE a − pE

=V[ =(

ecU qE + b − ] a − pE a − pE

a − pE ecU a − pE )V [ ( ) − 1] qE + b a − pE qE + b

= αV [

ecU a − pE ) − 1 ] (where α = qE + b qE + b

= αV [ U * − 1 ] . Therefore

dU dV = U ( 1 − V * ) and = αV [ U * − 1 ] . Q.E.D. dt dt

(*)

Michel 17 Bibliography

Allman, Elisabeth and John Rhodes. Mathematical Models in Biology. Cambridge UK 2004. Arrowsmith, D.K. and C.M. Place. Ordinary Differential Equations. Chapman and Hall New York New York, 1982. Britton, Nichoals. Essential Mathematical Biology. Springer London UK, 2003. Jones, D.S. and B.D. Sleeman. Differential Equations and Mathematical Biology. London UK 1983. Murray, James. Mathematical Biology. Springer-Verlag London UK, 1980. Michel, Aimee. Personal interview. 05 Aug. 2006. Nagle, Kent and Edward Saff. Fundamentals of Differential Equations. Sixth Edition. Pearson New York New York, 2004. Roberts, Fred. Discrete Mathematical Models with Applications in Biology. Englewood Cliffs NJ 1976. Venit, Stewart and Wayne Bishop. Elementary Linear Algebra. California State University at Los Angeles, CA 1996. Walpole, Ronald, Raymond and Sharon Myers and Keying Ye. Probability and Statistics for Engineers and Scientists. Upper Saddle River, NJ 2002.