On a geometric description of time dependent singular Lagrangians with applications to biological systems
aa r X i v : . [ q - b i o . P E ] A ug Singular Lagrangians and the dynamics of the ”Kill theWinner” model and its descendants
Sudip Garai ∗ and A Ghose-Choudhury † Department of Physics, Diamond Harbour Women’s University,D. H Road, Sarisha, West-Bengal 743368, IndiaPartha Guha ‡ SN Bose National Centre for Basic SciencesJD Block, Sector III, Salt LakeKolkata 700106, India
Abstract
We consider certain analytical features of the ”Kill the Winner” model which is astochastic model that can explain among other things competition among species and si-multaneous predation on the competing species. The model equations are shown to admita Jacobi Last Multiplier which in turns allows for the construction of a Lagrangian. TheLagrangian is of singular nature so that construction of the Hamiltonian via a Legendretransformation is not possible. A Hamiltonian description of the model therefore requiresthe introduction of Dirac brackets. Explicit results are presented for the model and itsreductions.
PACS:
Keywords:
Singular Lagrangians, Jacobi Last Multiplier, Dirac brackets, Kill the Winnermodel, Lotka-Volterra model, ∗ E-mail [email protected] † E-mail [email protected] ‡ E-mail: [email protected] Introduction
Population dynamics is a well established field and can lay claim to having increased ourunderstanding of several phenomena is diverse areas. However, most theoretical models ofpopulation dynamics do not offer a satisfactory explanation for the so-called diversity para-dox in nature. Roughly speaking when several species compete for the same finite resource, atheory called competitive exclusion indicates that one species will outperform the others anddrive them to extinction, thereby limiting biodiversity. However, nature does not quite evolvein this manner and shows up a greater degree of diversity then predicted. Goldenfeld and Xuedeveloped a stochastic model [19] that accounts for multiple factors observed in ecosystems,including competition among species and simultaneous predation on the competing species.Using bacteria and their host-specific viruses as an example, they showed that as the bacteriaevolve defenses against the virus, the virus population also evolves to combat the bacteria.This ”arms race” leads to a diverse population of both and to boom-bust cycles when a partic-ular species dominates the ecosystem then collapses- the so-called ”Kill the Winner” (KtW)phenomenon. This coevolutionary arms race is sufficient to yield a possible solution to thediversity paradox.It is of interest to look at such systems from a purely dynamic point of view as they provideinteresting concrete examples of first-order differential systems which have physical relevance.Of late there has been a lot of interest in the study of systems with singular Lagrangianswhich have arisen in the field of quantitative biology. Several well known models such as theLotka-Volterra prey-predator system, the host-parasite model, the Bailey model etc. are theexamples of systems that are described by singular Lagrangians, i.e. by Lagrangians that donot depend quadratically on the velocity parameter. A major problem in describing the dy-namics of such systems on the phase space is in defining the appropriate canonical momentum.In order to analyze the dynamics of such systems, Dirac formulated a method which basicallymakes the use of a constrained surface on which the Poisson bracket may be suitably defined.Roughly speaking this restricted Poisson bracket defined on a sub-manifold of the completephase space of the system is usually referred to the Dirac bracket.A Lagrangian is said to be singular if the corresponding Hessian matrix is singular.The singular nature of the Hessian matrix prevents us from solving for the velocities in termsof the conjugate momenta and the coordinates. This in turn prevents us from constructingthe Hamiltonian by means of a Legendre transformation. Most textbooks, while mentioningthis particular feature do not generally provide too many illustrative examples of singularLagrangians. Two illustrations are provided below. L = 12 m ( ˙ q + ˙ q ) + 12 µ ˙ q + V ( q , q , q ) L = ( q ˙ q + ˙ q ˙ q ) + ( q + q ) In each of the above cases it will be observed that the Hessian matrix (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ L∂ ˙ q i ∂ ˙ q j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) has a vanishing2eterminant. In the first instance the conjugate momenta are given by p = 1 m ( ˙ q + ˙ q ) , p = 1 m ( ˙ q + ˙ q ) , p = 1 µ ˙ q . As we can not solve for the ˙ q i in terms of the coordinates and momenta we are forced to treatthe above relations as a set of constraints, the so called primary constraints. We denote themby φ = p − m ( ˙ q + ˙ q ) ≈ φ = p − m ( ˙ q + ˙ q ) ≈ φ = p − µ ˙ q ≈ H = p ˙ q + p ˙ q + p ˙ q − L is therefore a function of the ”mixed” set of variables ( q i , ˙ q i , p i ). To develope a Hamiltoniandescription of the evolution of the systems we define the primary Hamiltonian as H p = H + X i λ i φ i Because the primary constraints must hold good at all times it is necessary that˙ φ i = { φ i , H p } = 0 , ∀ i = 1 , , . It is evident that if the Lagrangian is linear in the velocities then it is singular incharacter because the Hessian matrix is trivial. As already mentioned a particularly rich fieldin which the dynamics may be described by singular Lagrangians arises in models describingecological systems, predator-prey models etc.In this article we focus on an interesting stochastic model dealing with the problemof biological diversity. The so called ”Kill the Winner” (KtW) hypothesis has been quiteextensively studied by many authors. The present analysis is based on the model equationsintroduced in [19]. While most studies of such systems have relied on a numerical approach wehave adopted a more analytical view expanding on our previous work with biological systems.Several well known systems such as the Lotka-Volterra, Host-parasite, Kermack-McKendrickmodels may be shown to be special cases of the general KtW model studied here. It isinteresting to note that the stochastic model equations admit a Jacobi Last Multiplier (JLM)which in turn enables us to set up a Lagrangian description of the system. However, it isfound that the Lagrangian obtained is singular in character so that a proper Hamiltoniandescription of the system in phase space requires the introduction of Dirac brackets.Our main result is the following: 3 roposition 1.1
The generalized “Kill the Winner” model is described by the following sys-tem of ODEs ˙ x i = b i x i − m X j =1 e ij x i x j − p i x i y i , ˙ y i = q i x i y i − d i y i , i = 1 , ..., m where, a i , b i , c i are constants.(A) The Jacobi Last Multiplier for the system is given by M = Q mi =1 M i , with M i = e γ i t y σ i i /x i , γ i = e ii d i /q i and σ i = − e ii /q i .(B) The Lagrangian and Hamiltonian are singular, these are given by L = m X k =1 e γ k t y ekkqk k q k e kk ˙ x k x k − log x k ˙ y k y k + q k x k − d k log x k + 2 q k e kk m X j =1 ,j = k e kj x j − b k +2 p k y k (cid:18) q k e kk + 1 (cid:19)(cid:27)(cid:21) and H = − m X k =1 e γ k t y ekkqk k q k x k − d k log y k + 2 q k e kk m X j =1 ,j = k e kj x j − b k + 2 p k y k (cid:18) q k e kk + 1 (cid:19) respectively. By computing the Dirac brackets associated to this Hamiltonian we establish the requiredequations of motion of the phase space variables.The paper is organized as follows: In section 2 we outline the procedure for deriving aLagrangian linear in the velocities (and hence singular) assuming the existence of a JLM. Ageometric description of such singular Lagrangians is then presented followed by an introduc-tion to the notion of Dirac brackets. In section 3 we introduce the ”Kill the Winner” (KtW)model and considering its simplest version derive the explicit expression for the JLM. This isdone to motivate the subsequent general results. The lagrangian for the general KtW modelis then derived and the ingredients required for constructing the Dirac brackets is presented.This is followed by an explicit calculation of the Dirac brackets for some specific cases suchas the Lotka-Volterra model with and without competition and the Gierer-Meinhardt modelof pattern formation.
Let us briefly recall the procedure described in [12] for finding Lagrangians for a planar systemof ODEs from a knowledge of the last multiplier. We assume that the system dxdt = f ( t, x, y ) (2.1)4 ydt = g ( t, x, y ) (2.2)admits a Lagrangian which is linear in the velocities, so that L ( t, x, y, ˙ x, ˙ y ) = F ( t, x, y ) ˙ x + G ( t, x, y ) ˙ y − V ( t, x, y ) . (2.3)Then the Euler-Lagrange equations of motion ddt (cid:18) ∂L∂ ˙ u (cid:19) = ∂L∂u , with u = x and y yields ˙ y = (cid:18) F t + V x G x − F y (cid:19) = g ( t, x, y ) , (2.4)and ˙ x = − (cid:18) G t + V y G x − F y (cid:19) = f ( t, x, y ) , (2.5)where the subscripts on F, G and V denote the partial derivatives while the over-dots representthe derivative with respect to time. It is obvious that one must have G x = F y . In order tointroduce the notion of Jacobi’s last multiplier we assume that G x = − F y and assign a commonvalue, µ ( t, x, y ) := G x = − F y . (2.6)From (2.4) and (2.5), we have 2 µf ( t, x, y ) = − ( G t + V y ) (2.7)2 µg ( t, x, y ) = ( F t + V x ) . (2.8)It is clear that the construction ∂∂x (2 µf ) + ∂∂y (2 µg )leads to the following equation, ddt log µ + ∂f∂x + ∂g∂y = 0 . (2.9)using the original system of ODEs (2.1)-(2.2). However, (2.9) is precisely the defining relationfor JLM [18]. Thus we see that given the solution of this equation one can easily constructfrom (2.4) the coefficient functions F and G occurring in the expression for the Lagrangiansince F ( t, x, y ) = − Z µ ( t, x, y ) dy and G ( t, x, y ) = Z µ ( t, x, y ) dx. (2.10)5nce these functions are determined one can obtain an expression for the partial derivativesof V from (2.4) and (2.5) as follows ∂V∂x = 2 µ ( t, x, y ) g ( t, x, y ) + ∂∂t (cid:18)Z µdy (cid:19) , (2.11) ∂V∂y = − µ ( t, x, y ) f ( t, x, y ) − ∂∂t (cid:18)Z µdx (cid:19) . (2.12)In view of (2.9) it is easy to check the equality of the mixed derivatives, ∂ V∂x∂y = ∂ V∂y∂x .
The time evolution associated to a time-dependent mechanical system is usually representedby the flow of a vector field defined in R × T Q , where Q is a smooth differentiable manifoldwhich represents the original configuration space of the system. This is a space of 1-jets ofthe trivial bundle π : R × Q → R , i.e., J π = R × T Q . In local coordinates ( t, q i , v i ) in J π the local 1-forms θ i = dq i − v i dt , i = 1 , · · · , n , constitute a local basis for the contact 1-forms.The k -jet bundle of π is J k π = R × T k Q and there exists a natural projection π k,l : R × T k Q → R × T l Q for each pair of indices k, l and k > l . In the Lagrangian formalism, thedynamics takes place in the manifold R × T Q (for example, see for details [3, 16]).We briefly introduce the basic definition of vertical endomorphism this will is to definethe Poincar´e-Cartan form. Let v = v i ∂∂q i ∈ T q Q and V = ˙ q i ∂∂q i + ˙ v i ∂∂v i ∈ T v ( T Q ) such that τ T Q ( V ) = v . In local coordinates, τ T Q ( q i , v i , ˙ q i , ˙ v i ) = ( q i , v i ). If τ Q : v ∈ T Q q ∈ Q denotesthe natural projection, then, given a tangent vector V ∈ T v ( T Q ), we have τ T Q ( V ) = v . Let v ∈ T q Q be a vector tangent to Q at some point q ∈ Q . Then the vertical lift of v at a point w ∈ T q Q is the tangent vector v Vw ∈ T w ( T Q ) given by u Vw ( g ) = ddt g ( w + tv ) | t =0 ∀ g ∈ C ∞ ( T q Q ) . In local coordinates of T Q is given by v Vw = v i ∂∂v i | w , which is the Liouville vector field over T Q . The vertical endomorphism is the linear map S : T Q → T Q for which any vector V ∈ T Q yields S ( V ) = (( T u τ Q )( V )) V , where v = τ T Q ( V ) ∈ T Q .We can extend this theory of vertical endomorphism to the nonautonomous case. Thenin fibred coordinates, ( t, q i , v i ) ∈ R × T Q , S is given as S = ( dq i − v i dt ) ⊗ ∂/∂v i . The mostimportant objects associated to the time-dependent Lagrangian are the Poincar´e-Cartan oneand two forms defined by Θ L = dL ◦ S + Ldt, Ω L := − d Θ L . (2.13)6he local expressions areΘ L = ∂L∂v i ( dq i − v i dt ) + Ldt = (cid:0) L − v i ∂L∂v i (cid:1) dt + ∂L∂v i dq i (2.14)Ω L = − d (cid:0) ∂L∂v i (cid:1) ∧ dq i + d (cid:0) v i ∂L∂v i − L (cid:1) ∧ dt. (2.15) Let M be a 2 n -dimensional symplectic manifold, this can be manifested as a classical phasespace of a system with n -degrees of freedom. Dirac bracket were introduced as a modificationof Poisson bracket in presence of constraints. If we impose 2 m independent constraints, thisyields 2( n − m )-dimensional symplectic manifold M c ⊂ M . In the neighbourhood of a point p ∈ M , choose coordinates x , · · · x n ∈ M , such that M c is given by x = 0 , · · · , x m − =0 , x m = 0 , constraints are forcing the system to lie on M c . Instead of functions, we can supply2( n − m ) vector fields whose kernel is M c . Thus x m +1 , · · · , x n provide local coordinates on M c . Let us define the matrix C rs ( x ) = { x r , x s } , r, s = 1 , · · · m, and suppose f, g ∈ C ∞ ( M ) be the smooth functions on M , and let f ′ , g ′ be their restrictionto M c . Definition 2.1
A function f ∈ C ∞ ( M ) is called first class if { f, x r } = 0 , ∀ r , where x r ( q, p ) =0 , otherwise it is called second class. All constraints are called second class provided { x r , x s } = C rs , and all constraints are called first class if { y r , y s } = f wrs y w ≈ . Then the Dirac bracket is defined by { f ′ , g ′ } D := { f, g } − m X r,s =1 { f, x s } [ C rs ] − { x s , g } , (2.16)where the double sum is taken for all second class constraints. For any second class constraint { f ′ , x k } D := { f, x k } − m X r,s =1 { f, x r } [ C rs ] − { x s , x k }{ f, x s } = { f, x k } − m X r,s =1 { f, x r } [ C rs ] − [ C sk ]= { f, x k } − m X r =1 { f, x r } δ rk = { f, x k } − { f, x k } = 0 . D := Π + 12 [ C rs ] − X r ∧ X s , (2.17)where Π stands for Poisson bivector and X r = {· , x r } = − ( ∂ i x r )Π ij ∂ j . Let { p j , q j } , 1 ≤ j ≤ n , denote a set of dynamical variables, { u a } , 1 ≤ a ≤ m , set ofLagrange multipliers, and { x a ( p, q ) } a set of constraints. The total Hamiltonian is given by H T ot = H ( p, q ) + u a x a ( p, q ) . (2.18)Then the dynamics of a constrained system can be obtained from the action principle S = Z [ p j ˙ q j − H ( p, q ) − u a x a ( p, q )] dt. The resultant equations that arise from the action read˙ p i = − ∂H∂q i + u a ∂x a ∂q i , ˙ q i = ∂H∂p i − u a ∂x a ∂p i , ˙ x a = 0 . (2.19)The (constrained) Hamiltonian equation for any arbitrary smooth function g ( p, q ) ∈ C ∞ ( M )is given by˙ g = { g, H ( p, q ) + u a x a } = { g, H ( p, q ) } + { g, u a x a } ≈ { g, H ( p, q ) } + u a { g, x a } , (2.20)where the ≈ symbol means that we should not substitute x a = 0 before evaluating Pois-son bracket. This is called the weak equation. For a variation of the weak equation ˙ g ≈{ g, H ( p, q ) + u a x a } , the variation of LHS is not equal to the variation of RHS.Let us assume that all the constraints are second class, then the corresponding unde-termined coefficients are given by u b = −{ H, x a } [ C ab ] − . (2.21)We can express the equation of motion with second class constraints in terms of Dirac bracket. Proposition 2.1
The equation of motion for a system with second class constraints can beexpressed by ˙ g ≈ { g, H } D = { g, H } − X a,b { g, x a } [ C ab ] − { x b , H } . (2.22)This can be proved easily using (2.21). 8 Singular Lagrangians of biological systems and Diracbracket
The “kill the winner (KtW)” hypothesis which attempts to get a stochastic model for thebiological diversity problem in nature where the host-specific predators control the prey pop-ulation of each species by preventing a winner from emerging. This natural phenomenonmaintains the equilibrium coexistence of all the existing species in the system. An individual-level stochastic model in which predator-prey coevolution promotes the high diversity of theecosystem by generating a persistent population flux of species has been developed in [19].For a single species the model is described by the following system of differential equations[19]: ˙ x = a x − b x − c xy (3.1)˙ y = a xy − b y (3.2)where, a , a , b , b , c are the constants and x ( y ) represents the bacterial(viral strains) densityof the system. The Jacobi last multiplier for this system can be written as M = e γt x α y β with α = − β = b − a a (3.4) γ = b b a (3.5)The generalized KTW model is given by the system of equations˙ x i = b i x i − m X j =1 e ij x i x j − p i x i y i (3.6)˙ y i = q i x i y i − d i y i , i = 1 , ..., m (3.7)The Jacobi last multiplier for the system is given by M = m Y i =1 M i (3.8)with M i = e γ i t y σ i i /x i . Here γ i = e ii d i /q i and σ i = − e ii /q i . Now assume the singularLagrangian as L = m X k =1 [ F k ( x, y ) ˙ x k + G k ( x, y ) ˙ y ] − U ( x, y ) (3.9)substituting into the Euler-Lagrange equation it gives m X k =1 [( F ix k − F kx i ) ˙ x + ( F iy k − G kx i ) ˙ y ] = − (cid:18) ∂U∂x i + ∂F i ∂t (cid:19) (3.10)9 X k =1 [( G ix k − F ky i ) ˙ x + ( G iy k − G ky i ) ˙ y ] = − (cid:18) ∂U∂y i + ∂G i ∂t (cid:19) (3.11)This can be written in a matrix form as (cid:18) ˙ X ˙ Y (cid:19) = − (cid:18) A B − B T D (cid:19) − (cid:18) PQ (cid:19) = − (cid:18) D − A − BAD − B T D A (cid:19) (cid:18) PQ (cid:19) where X = x ... x m , Y = y ... y m , P = ∂F ∂t + ∂U∂x ... ∂F m ∂t + ∂U∂x m , Q = ∂G ∂t + ∂U∂y ... ∂G m ∂t + ∂U∂y m Now we make the assumption that A = B = 0 and B is diagonal with B ii = F iy i − G ix i .Therefore we can write ˙ y i = − (cid:16) ∂U∂x i + ∂F i ∂t (cid:17)(cid:16) ∂F i ∂y i − ∂G i ∂x i (cid:17) (3.12)˙ x i = (cid:16) ∂U∂y i + ∂G i ∂t (cid:17)(cid:16) ∂F i ∂y i − ∂G i ∂x i (cid:17) (3.13)Next let us assume, ∂F i ∂y i = − ∂G i ∂x i = M i (3.14)therefore, ˙ y i = − M i (cid:18) ∂U∂x i + ∂F i ∂t (cid:19) (3.15)& ˙ x i = − M i (cid:18) ∂U∂y i + ∂G i ∂t (cid:19) (3.16)It follows that F i = Z M i dy i + f i ( t, x i ) = e γ i t x i y σ i +1 i ( σ i + 1) + f i ( t, x i ) (3.17)and G i = − Z M i dx i + g i ( t, y i ) = − e γ i t y σ i i log x i + g i ( t, y i ) (3.18)Now from the above relations we can see that ∂U∂x i = − e γ i t y σ i +1 i x i (cid:20) q i x i − d i ) + γ i ( σ i + 1) (cid:21) (3.19)10nd ∂U∂y i = e γ i t y σ i i " b i − m X j =1 e ij x j − p i y i + γ i log x i (3.20)The solution for the potential U can be obtained from the above set of equations as U = − m X i =1 e γ i t y σ i +1 i (cid:26) q i x i − d i log x i + 2 p i y i ( σ i + 2) (cid:27) + e γ i t y σ i +1 i q i e ii m X j =1 ,j = i e ij x j − b i (3.21)Hence the Lagrangian for the system can be written as L = m X k =1 e γ k t y ekkqk k q k e kk ˙ x k x k − log x k ˙ y k y k + q k x k − d k log x k + 2 q k e kk m X j =1 ,j = k e kj x j − b k +2 p k y k (cid:18) q k e kk + 1 (cid:19)(cid:27)(cid:21) (3.22)The expression for the Hamiltonian turns out to be H = m X k =1 (cid:20) ∂L∂ ˙ x k ˙ x k + ∂L∂ ˙ y k ˙ y k (cid:21) − L = U ( x, y )= − m X k =1 e γ k t y ekkqk k q k x k − d k log y k + 2 q k e kk m X j =1 ,j = k e kj x j − b k + 2 p k y k (cid:18) q k e kk + 1 (cid:19) and is independent of the velocities as is natural for such singular Lagrangians. This completesthe proof of the proposition (1.1).Let us now identify the primary constraints of the system: It is easily seen that theconjugate momenta are given by p xk = ∂L∂ ˙ x k = F k = e γ k t q k e kk y e kk /q k k x k p yk = ∂L∂ ˙ y k = G k = − e γ k t y e kk /q k − k log x k These lead us to the primary constraints of the model which are defined as φ k := p xk − F k ≈ , ψ k := p yk − G k ≈ . (3.23)The primary Hamiltonian is therefore H p = H + m X k =1 ( λ k φ k + µ k ψ k ) , k = 1 , ..., m (3.24)11s the primary constraints must be satisfied at all times it is necessary that their time evolutionvanish. This requirement leads us to the second class constraints of the system.˙ φ k = { φ k , H } + m X k =1 µ k { φ x , ψ k } ≈ φ -class of primary con-straints by φ k + m = { p xk , U } + m X k =1 µ k { φ k , ψ k } ≈ , k = 1 , ..., m (3.25)In a similar manner for the ψ -class of primary constraints one defines the correspondingsecondary constraints as ψ k + m = { p yk , U } + m X k =1 λ k { ψ k , φ k } ≈ , k = 1 , ..., m (3.26)Let us now define the matrix of the Poisson brackets between the all the primary andsecond class constraints as C which is obviously a skew symmetric 4 m × m matrix. C = (cid:20) { φ i , φ j } m × m { φ i , ψ j } m × m { ψ i , φ j } m × m { ψ i , ψ j } m × m (cid:21) This turns out to be non-singular and in terms of its elements the time evolution of anyvariable f is given by˙ f = { f, H } − ( { f, φ } , · · · , { f, φ m } , { f, ψ } , · · · { f, ψ m } ) C − { φ , H } ... { φ m , H }{ ψ , H } ... { ψ m , H } (3.27)we now present the explicit nature of the calculations stated above by considering the case m = 1, i.e, for the system (3.1) and (3.2). The JLM it will be recalled is given by J = e γt y σ /x where γ = e d/q and σ + 1 = e /q . It follows that the Lagrangian for the system is given by L = e γt y σ +1 x ( σ + 1) ˙ x − ( e γt y σ log x ) ˙ y − U ( x, y, t )where U ( x, y, t ) = − e γt y σ +1 (cid:20) pσ + 2 y + 2 qx − d log x − qb e (cid:21) H = U and the primary constraint equations whichfollow from the usual definition of the conjugate momenta are φ = p x − e γt y σ +1 x ( σ + 1) ≈ φ = p y + e γt y σ log x ≈ H p = U ( x, y, t ) + λ φ + λ φ and from the time evolution of the primary constraints we arrive at the following second-classconstraints, namely: φ = − U x − λ (cid:18) e γt y σ x (cid:19) ≈ φ = − U y + λ (cid:18) e γt y σ x (cid:19) ≈ C = − e γt y σ x − φ x − φ x e γt y σ x − φ y − φ y φ x φ y φ x φ y and its inverse is given by C − = 1∆ φ y − φ y − φ x φ x − φ y φ x − e γt y σ x φ y − φ x e γt y σ x where ∆ = φ x φ y − φ x φ y . Explicit calculation of the time evolution of the phase spacevariables ( x, y, p x , p y ) using the Dirac brackets now yields˙ x = 0 , (3.28)˙ y = 0 , (3.29)˙ p x = ( − U x , (3.30)˙ p y = ( − U y . (3.31)We now turn to certain simplified reductions of the KtW model presented above and deduceexplicitly the equations of motion for the phase space variables.13 .1.1 Lotka-Volterra model with competition This is a model similar to the original Lotka-Volterra model but which incorporates the compe-tition between species which is modeled by a term proportion to the product of the populationsof the prey and predator. The equations are given by˙ x = x ( a − b x − c y ) (3.32)˙ y = y ( a − b y − c x ) (3.33)where a i , b i , c i > ∀ i = 1 ,
2. It is evident by comparison with (3.1) and (3.2) that thismodel is a special case of the KtW model with an extra term proportional to y in the secondequation. The Jacobi Last Multiplier for this system of equations is given by µ = e γt x α y β with the exponents being α = b c + c c − b b b b − c c β = b c + c c − b b b b − c c γ = a ( b b − b c ) + a ( b b − b c ) b b − c c It turns out that a singular Lagrangian for the above system is given by L = − e γt x α y β +1 β + 1 ˙ x + e γt x α +1 y β α + 1 ˙ y − V ( x, y, t )where V ( x, y, t ) = e γt x α +1 y β +1 (cid:20) a − b y ) α + 1 + γ ( α + 1)( β + 1) − c xα + 2 (cid:21) or alternately = e γt x α +1 y β +1 (cid:20) − a − b x ) β + 1 − γ ( α + 1)( β + 1) + 2 c yβ + 2 (cid:21) As characteristic feature of singular Lagrangians is that the Hamiltonian is given by H = p x ˙ x + p y ˙ y − L = V ( x, y, t )and is therefore independent of the velocities. The primary constraints are therefore φ = p x − F ( x, y, t ) ≈ φ = p y − G ( x, y, t ) ≈ F ( x, y, t ) = − e γt x α y β +1 β + 114 ( x, y, t ) = e γt x α +1 y β α + 1respectively. Hence the primary Hamiltonian is H p = V + λ φ + λ φ . On the other hand the second class constraints are given by φ = ˙ φ = { φ , H p } = { p x , V ( x, y, t ) } + λ { φ , φ } ≈ φ = ˙ φ = { φ , H p } = { p y , V ( x, y, t ) } + λ { φ , φ } ≈ φ = − V x + λ (2 e γt x α y β ) ≈ φ = − V y − λ (2 e γt x α y β ) ≈ C = e γt x α y β V xx − αV x /x V xy + αV x /x − e γt x α y β V xy + βV y /y V yy − βV y /y − V xx + αV x /x − V xy − βV y /y − V xy − αV x /x − V yy + βV y /y This is a non-singular matrix and its inverse is C − = 1∆ − V yy + βV y /y V xy + βV y /y V xy + αV x /x − V xx + αV x /xV yy − βV y /y − V xy − αV x /x e γt x α y β − V xy − βV y /y V xx − αV x /x − e γt x α y β where ∆ = ( V xx V yy − V xy ) − αV x ( V xy + V yy ) /x − βV y ( V xy + V xx ) /y . The time evolution of adynamical variable f is in terms of the Dirac brackets˙ f = { f, H } DB = { f, H } − { f, φ i } [ C − ] ij { φ j , H } and up on using the entries of the matrix C − as given above we find that˙ x = 0 , ˙ y = 0 , ˙ p x = − V x − (cid:20) ( V xx − αV x x ) { ( − V yy + βV y y ) V x + ( V xy + αV x x ) V y } +( V xy − αV y x ) { ( − V xy + βV y y ) V x + ( − V xx + αV x x ) V y } (cid:21) ˙ p y = − V y − (cid:20) ( V xy − βV x y ) { ( − V yy + βV y y ) V x + ( V xy + αV x x ) V y } +( V yy − βV y y ) { ( V xy + βV y y ) V x + ( − V xx + αV x x ) V y } (cid:21) .1.2 Lotka-Volterra model without competition In the absence of competition, the coefficients ( b , b ) in the Lotka-Volterra model with com-petition vanish and a simplified version of the predator-prey systems becomes˙ x = ax − bxy (3.34)˙ y = − cy + dxy (3.35)where x and y refer to two species which live in a limited area with individual of the species y (predator) feed only on the species x (prey). The parameters a, b, c and d are assumed tobe positive. The overdots refer to derivatives with respect to time with ˙ x representing thegrowth rate of the prey population and ˙ y being the growth rate of the predator population.This model can also be visualized as the special case of the KtW model with the absence ofthe term proportional to x in the ˙ x equation.One can recast the system (3.34) and (3.35) in the form of Euler-Lagrange equation withthe Lagrangian L = (cid:18) − log xy ˙ x + log yx ˙ y (cid:19) − c log x + a log y + dx − by, (3.36)and the corresponding momenta are p x = ∂L∂ ˙ x = − log yx , p y = ∂L∂ ˙ y = log xy . (3.37)The Hamiltonian is therefore given by H = − c log x − a log y + 2 dx + 2 by. (3.38)The Hamiltonian is clearly independent of the momentum (or velocity) and is therefore ofsingular character. In order to employ study the dynamics in the phase space ( x, y, p x , p y ) wetreat the momenta as given in (3.37) as the primary constraints and express these in the form φ = p x + log yx ≈ , φ = p y − log xy ≈ H p = H + λ φ + λ φ (3.40)where λ and λ are Lagrange multipliers. In order to ensure that the constraints hold at alltimes it is necessary that their time evolutions with respect to the Hamiltonian H p vanish. Inother words we require that ˙ φ = { φ , H } = 2 cx − d + 2 λ xy ≈ φ = { φ , H } = 2 ay − b − λ xy ≈ φ = 2 cx − d + 2 λ xy ≈ , φ = 2 ay − b − λ xy ≈ C denote that matrix formed by the Poisson bracketsbetween φ i and φ j , i.e, C ij = { φ i , φ j } . The ≈ sign here means we can only substitute thevalue of λ i from the secondary constraint equations after working out the respective Poissonbrackets. With this in mind it is found that C = 2 xy dy − ( a − by ) − − ( c − dx ) bx − dy ( c − dx ) 0 0( a − by ) − bx (3.42)It may be verified that C is non-singular and its inverse is given by C − = 2 xyξ − bx − ( c − dx )0 0 − ( a − by ) − dybx ( a − by ) 0 1( c − dx ) dy − (3.43)where det C = ξ = adx + bcy − ac . The time evolution of any dynamical quantity in terms ofthe Dirac bracket is defined as˙ f = { f, H } DB = { f, H } − { f, φ i } [ C − ] ij { φ j , H } (3.44)Using this definition we find in the present case the following equations of motion.˙ x = 0 , ˙ p x = 2( c − dx ) x (cid:18) − x y (cid:19) (3.45)˙ y = 0 , ˙ p y = 2( a − by ) y (cid:18) − x y (cid:19) (3.46) A simplified version of this model which is often cited in the literature is given by the followingsystem of differential equations: ˙ x = − k xy (3.47)˙ y = k xy − k y (3.48)17ith k and k being positive constants. This model can also be derived from the KtWmodel by suppressing some coefficients. The system may be derived from the Euler-lagrangeequations with the Lagrangian L = 12 (cid:18) log yx ˙ x − log xy ˙ y (cid:19) + k ( x + y ) − k log x (3.49)The conjugate momenta are obtained from p x = ∂L∂ ˙ x = log y x ,p y = ∂L∂ ˙ y = − log x y , and are obviously velocity independent as the Lagrangian is singular in character. This ne-cessitates that we defined the primary constraints by φ = p x − log y x ≈ , φ = p y + log x y ≈ . (3.50)The Hamiltonian therefore has the appearance H = p x ˙ x + p y ˙ y − L = − k ( x + y ) + k log x. (3.51)It is easy to verify that H is a constant of motion. We define the primary Hamiltonian by H p = H + λ φ + λ φ . (3.52)In order to ensure that the primary constraints hold at all times it is necessary that ˙ φ i = { φ i , H p } i = 1 , φ = { φ , H p } = { φ , H } + λ { φ , φ } whence we have φ = k − k x − λ xy ≈ . (3.53)Similarly considering the vanishing of the time evolution of G we find that φ = k + λ xy ≈ . (3.54)The matrix of the Poisson brackets of the primary and secondary constraints is thengiven by C = − xy − k x − k x xy − k y + k xy − k yk x k y − k xy k x k y (3.55)18his is a non-singular matrix and its inverse is given by C − = xk k k x − k x + k − k y k y − k x k y − k x − k − k y (3.56)If we go back to the definition of the Dirac bracket it will be realized that the equationsof motion of the phase space variables are now obtained from˙ f = { f, H } DB = { f, H } − k k ( { f, φ } , { f, φ } , { f, φ } , { f, φ } ) − k x ( x − y ) + k k x ( k x − k ) − k xy It may be verified that this implies ˙ x = 0˙ y = 0˙ p x = ( k k − (cid:18) − k + k x (cid:19) ˙ p y = k − k k This is a reaction-diffusion model which deals with the formation of patterns. A simplifiedversion of this model neglecting the effects of diffusion is given by˙ x = x y − ax (3.57)˙ y = b − x y − y. (3.58)In order to express this system in the form of Euler-Lagrangian equations, it is found thatunless the parameter b = 0, we cannot find a Lagrangian. Simple calculations show that with b = 0 one can derive a Jacobi Last Multiplier for this system which is given by µ = e − at x y and the Lagrangian (singular) is of the form L = e − at (cid:20) log yx ˙ x + 1 xy ˙ y − (cid:18) x − x ) + 2 y − a log yx (cid:19)(cid:21) The Hamiltonian is then given by H = e − at (cid:20) x − x ) + 2 y − a log yx (cid:21) φ = p x − e − at log yx ≈ φ = p y − e − at xy ≈ H p = H + λ φ + λ φ By demanding that the time evolution of the primary constraints vanish we are led to thefollowing secondary constraints, as already explained earlier, namely φ = e − at (cid:18) x + ax log y + 2 λ x y (cid:19) ≈ φ = e − at (cid:18) axy − λ x y (cid:19) ≈ C = 2 e − at x y − − xy xy − a ǫ x xy − ǫ − xy + a − x where ǫ = − ( x + 1 + a a y )Straightforward calculations give the inverse of the matrix M to be C − = e at x y δ − x ǫ − xy − a xyx − xy + a − − ǫ − xy where δ = 2 x y − ǫ (2 xy − a Conclusion
In the diverse field of microbial systems where the mutation rates of different species arevery high and also in the field of quantitative biology like prey-predator system, host-parasitemodel etc. where the prey predator relationships becomes much more complicated with re-spect to the time parameter exact analytical results are often rare. In this paper we haveattempted to provide a Lagrange/Hamiltonian description of an important stochastic model,popularly referred to as the ”Kill the Winner” model. It is evident thuat6 such a descriptionis facilitated by the existence of a Jacobi Last Multiplier which is not very well known outsidethe community of mathematicians working on systems of ordinary differential equations. Itsexistence leads quite naturally to a Lagrange description of the model equations, which in thepresent case turns out to be of a singular character. As a result the corresponding Hamilto-nian description of the phase space variables requires the intro duction of Dirac brackets. Wehave presented explicit results for a number of models such a predator-prey systems with andwithout competition, pattern formation equations and of course the KtW model which formsthe cornerstone of the article.
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