Stochastic Differential Equations in Population Dynamics

Transcription

1 Stochastic Differential Equations in Population Dynamics Numerical Analysis, Stability and Theoretical Perspectives Bhaskar Ramasubramanian Abstract Population dynamics in the presence of noise in the environment can be modeled reasonably well by stochastic differential equations. The one dimensional logistic equation is analyzed by the Runge-Kutta and Euler-Runge-Kutta methods. An error analysis is performed and the order of convergence of these methods is determined experimentally. This model is extended to the two dimensional case and instances of interactions among individuals of the same and different species is separately considered - the former is presented through a model for epidemics while the latter is presented via a predator-prey model. The stabilizing effect of noise in these models is examined. Finally, results of the simulation exercises are compared with theoretical results to complete the picture. 1 Introduction This report studies stochastic differential equation (SDE) models of population dynamics. The next section introduces the logistic equation in one and two dimensions. The stochastic logistic equation in one dimension is the subject of a detailed error analysis using three different numerical algorithms. Though this model serves as a good starting point, more interesting information can be elicited by studying interactions among different subsets of the population. This leads to the two dimensional logistic equation. The third section gives an insight into the modeling of diseases and epidemics using differential equations. The deterministic case is presented first and the stochastic framework is built upon this. A small digression is made to make the reader aware of constructing an epidemic model via a Birth-Death process (naturally leading to a convenient Markov Chain formulation). Two possibilities are examined - the first, when an infected individual, once cured, does not become immune to the disease (SIS), and the second, when there is a non-zero probability of an infected individual becoming immune or removed from the disease (SIR). The fourth section presents a predator-prey model where populations of different species interact with each other. Like in the section preceding it, the deterministic model serves as the building block for the stochastic case. The effect of environmental noise on the size and persistence time of the population is studied. While the one dimensional logistic equation makes itself amenable to analysis by having a closed form solution, the same problem in higher dimensions does not have a closed form solution. Numerical techniques, in this case, serve as an effective medium to study the (asymptotic) effects of noise on the dynamics. Since the addition of noise alters the dynamics of the deterministic model, it is not surprising that certain asymptotic properties are also different between the two cases. The effect on the stability of the deterministic model when noise is added to it is studied in some detail. The numerical analysis of the models presented naturally has an underlying theoretical foundation. The fifth section aims to establish a connection between the results of the simulation exercises and theory. Experimental results and observations are presented at the end of each section, wherever applicable. Primary referenc(es) for each (sub)section have been cited at the beginning of the respective (sub)section. Other references, if used in the same (sub)section, have been cited at the appropriate instances. All simulations have been carried out in MATLAB R. 1

2 2 The Logistic Equation D Case ([4]) The one dimensional logistic equation, also called the Verhulst equation, models the rate of growth of a single species of population. The underlying principle is that while the unimpeded growth of the population is positive, the rate of growth decreases as the population starts to compete (among themselves) for resources. Mathematically, the model can be written as : dx(t) = X(t)(λ X(t))dt (1) X(0) = X 0. λ is called the growth rate. In the stochastic setting, a noise term is added to this equation, which can then be written as : X(0) = X 0. The exact solution to equation (2) is given by : Algorithms Used for Error Analysis dx(t) = X(t)(λ X(t))dt + σx(t)dw t (2) X 0 exp((λ 0.5σ 2 )t + σw t ) X(t) = t 1 + X 0 0 exp((λ 0.5σ2 )s + σw s )ds Consider a stochastic differential equation in the following form : dx(t) = f(x)dt + g(x)dw t (4) The above equation is discretized by replacing dt by t i and dw t by W i. The following three algorithms were used for error analysis of the solution of the one dimensional logistic equation in this report. Note that x 0 = X 0 in each case, and the index i takes integer values, starting from 0. Euler-Maruyama Method Runge-Kutta Method x i+1 = x i + f(x i ) t i + g(x i ) W i (5) x i+1 = x i + f(x i ) t i + g(x i ) W i + 0.5(g(x i + g(x i ) t i ) g(x i )) W i 2 t i (6) ti Euler-Runge-Kutta Method k 0 0 = f(x i ) k 1 0 = g(x i ) k 0 1 = f(x i + k 0 0 t i + k 1 0δW i ) k 1 1 = g(x i 2 3 k0 0( W i 3 t i )) k 1 2 = g(x i 2 9 k0 0(3 W i + 3 t i )) k3 1 = g(x i k0 0 t i ((k1 1 k0) W 1 i k1 1 3 ti ) x i+1 = x i (k0 0 + k1) t 0 i (37k k2 1 27k3) W 1 i (8k1 0 + k1 1 9k2) 1 3 t i (7) The Runge-Kutta method is got by an approximation of the partial derivative term in the Millstein algorithm seen during the course. The Euler-Runge-Kutta procedure reminds one of the fourth order Runge-Kutta algorithm used for ordinary differential equations. Note that the Runge-Kutta and the Euler-Runge-Kutta methods do not require the explicit computation of the derivatives of the functions involved at every step, which make them attractive for numerical implementation. (3) 2

3 2.2 2-D Case ([6]) For dimensions higher than one, the logistic equation does not have a closed form solution. Hence, numerical techniques have to be resorted to for analyzing such systems. Consider the system : dx 1 (t) = X 1 (t)(b 1 a 11 X 1 (t) + a 12 X 2 (t))dt dx 2 (t) = X 2 (t)(b 2 a 22 X 2 (t) + a 21 X 1 (t))dt (8) with initial populations X 1 (0) = X 10 and X 2 (0) = X 20. This system is called a facultative mutualism, i.e. each species enhances the growth of the other although each can survive in the absence of the other. An example of this symbiotic behavior is seen in nature between sea anemones and hermit crabs. Each species plays a role in driving away possible predators of the other. Analogous to the one dimensional case, in the stochastic framework, the equation are : dx 1 (t) = X 1 (t)((b 1 a 11 X 1 (t) + a 12 X 2 (t))dt + (σ 11 X 1 (t) + σ 12 X 2 (t))dw 1 (t)) dx 2 (t) = X 2 (t)((b 2 a 22 X 2 (t) + a 21 X 1 (t))dt + (σ 21 X 1 (t) + σ 22 X 2 (t))dw 2 (t)) (9) with b i, a ij, σ 11, σ 22 > 0 and σ 12, σ Experimental Results and Observations The 1 D Case Figure 1: Trajectory for the Euler-Runge-Kutta Method 3

4 Figure 2: Error (Strong) Plot for Euler-Maruyama Method Figure 3: Error (Strong) Plot for Strong Runge-Kutta Method 4

5 Figure 4: Error (Strong) Plot for Euler-Runge-Kutta Method The true solution for the one dimensional logistic equation got from equation (3) was compared with a realization got from the Euler-Runge-Kutta algorithm with initial population X 0 = 1 and λ = σ = 1. This is shown in figure (1). The trajectories were found to agree and the error at the final time between the two trajectories was Figures (2, 3 and 4) respectively show the results of an error analysis performed on the Euler-Maruyama, Runge-Kutta and Euler-Runge-Kutta methods for 5000 realizations of the Brownian motion and five different time steps for each. For each sample path and step size, the error at the upper boundary of the interval [0, 1] was computed and the average over all samples for each step size was used to generate the error plot. The (strong) order of convergence was determined by comparing the error got from the numerical simulation (blue line) with a reference line of appropriate slope (red line) on a log-log plot ([2]). While the E-M algorithm had a strong error of order 0.5, the other two methods had strong orders 1, which agrees with the results presented in the literature. A peculiar observation for the strong error plot of the E-R-K method was that the error plot did not agree with the reference for the entire time period considered. The slope of the error plot is seen to suddenly increase at a certain point. One explanation for this could be the large number of intermediate steps involved in each iteration, leading to a possible accumulation of errors. As mentioned earlier, the Runge-Kutta method is derived by approximating the partial derivative term in the Milstein method ([2]), which means that the order of convergence of this method must match that of the Milstein algorithm. The residual errors in the three cases were respectively , and The residual error for the E-R-K algorithm is seen to be significantly higher than that for the other two methods, which confirms suspicions of errors accumulating due to the number of intermediate steps at each stage of the algorithm The 2 D Case The model considered for the experiment was : dx 1 (t) = X 1 (t)((1 X 1 (t) + 2X 2 (t))dt + (σx 1 (t))dw (t)) dx 2 (t) = X 2 (t)((1 2X 2 (t) + 2X 1 (t))dt + (2σX 2 (t))dw (t)) (10) 5

6 Realizations of X 1 (t) and X 2 (t) were plotted for a noise level σ = 1. The realizations were compared with the deterministic solution, i.e. by setting σ = 0 in equation (10). Figure 5: No. of X 1 : Deterministic vs. Stochastic Figure 6: No. of X 2 : Deterministic vs. Stochastic Figures (5) and (6) show that the noise plays a role in suppressing the explosion of the population ([6]). 6

7 The blue curve shows the deterministic case, when the population explodes to infinity in finite time, while the red curve shows the case when the system is randomly perturbed. The noise in the environment can thus be considered to have a stabilizing effect on the deterministic dynamics. This notion is formalized in the Section 5 of this report. 3 The Epidemic Model ([1]) 3.1 Birth-Death Processes A Markov process is a stochastic process in which the future state depends only on the present state and not how the current state was reached. The birth-death process is a special case of a (continuous time) Markov process in which there are only two types of state transitions - births, when the state increases by one and deaths, when the state decreases by one. Naturally, the states of the system are the number of people alive. Transitions among states in the Markov chain for a birth-death process occur only if the states are adjacent to each other, i.e. p i,j 0 if and only if j {i 1, i + 1}. Thus, the transition matrix is tridiagonal. 3.2 The SIS Model In this scenario, a susceptible individual (S), on contact with an infected individual (I), becomes infected and infectious, but does not develop immunity to the disease. Thus, after recovery, the individual returns to the susceptible class. The SIS model has been applied in the study of the spread of sexually transmitted diseases. The differential equations describing the model can be written as : Ṡ = β SI + (b + γ)i N I = β SI (b + γ)i (11) N where β > 0 is the contact rate between the two classes of the population, γ > 0 is the recovery rate (from the infected state), b 0 is the birth rate and N = S(t)+I(t) is the total population size. In the results presented in this report, the size of the population is assumed to remain constant, i.e. N = S(t) + I(t) = S(0) + I(0). The asymptotic dynamics are completely determined by the basic reproduction number, which is the number of secondary infections caused by one infected individual in an entirely susceptible population. In terms of the parameters of the dynamics, this is given by R 0 = β b+γ. Theorem 3.1 Let S(t) and I(t) be a solution to the model in (11). If R 0 1, then lim t (S(t), I(t)) = (N, 0) - disease free equilibrium. If R 0 > 1, then lim t (S(t), I(t)) = ( N R 0, N(1 1 R 0 )) - endemic equilibrium 3.3 The SIR Model In this model, infected individuals (I) can get completely immune to the disease and enter what is called a removed class (R). The SIR model has been used to study the spread of diseases like chickenpox and measles. The differential equations describing the model are : Ṡ = β SI + b(i + R) N I = β SI (b + γ)i N Ṙ = γi br (12) where β > 0, γ > 0, b 0 and N = S(t) + I(t) + R(t) = S(0) + I(0) + R(0). The initial conditions satisfy S(0) > 0, I(0) > 0 and R(0) 0. As in the SIS case, the total size of the population is assumed to remain constant. The following result summarizes the asymptotic dynamics of the system : 7

8 Theorem 3.2 Let S(t), I(t) and R(t) be a solution to the model in (12). S(0) R 0 N If R 0 1, then lim t I(t) = 0 - disease free equilibrium. If R 0 > 1, then lim t (S(t), I(t), R(t)) = ( N R 0, bn b+γ (1 1 R 0 ), γn b+γ (1 1 R 0 )) - endemic equilibrium. Assume b = 0. If R 0 S(0) N if R 0 S(0) N > 1, then there is an initial increase in the number of infected cases I(t), but 1, then I(t) decreases monotonically to zero. is called the initial replacement number, the average number of secondary infections produced by an infected individual during the period of being infected at the outset of the epidemic. Since this infectious fraction changes with time, the replacement number is defined as R 0 S(t) N. It will subsequently be seen that R 0 > 1 is the more interesting of the two cases, as the behaviors of the stochastic and deterministic models differ significantly under this condition. 3.4 Extension to the Stochastic Case The total population remaining constant, the SIS model has one independent random variable, I(t), while the SIR model has two independent random variables in {S(t), I(t)}. The addition of noise in this setting leads to the formulation of an SDE, whose solution will be a sample path of the stochastic process SDE for SIS Model The Euler-Maruyama method will be applied to the following SDE : di = µ(i)dt + σ(i)dw t Here, µ(i) and σ(i) are parameters corresponding to the instantaneous growth and variance of the birthdeath process. For the SIS case, this equation is : di = ( β N (N I)I (b + γ)i)dt + ( β N (N I)I + (b + γ)i)dw t (13) SDE for SIR Model The SDE for the SIR model will be presented with the assumption that there are no births, i.e. b = 0, to simplify derivations. Consider the tuple dx = (ds di) T. The expectation and covariance matrix of dx to order dt is : ( β E(dX(t)) = N SI β N SI γi ( β V ar(dx(t)) = N SI β N SI β N SI β N SI γi ) dt (14) ) dt (15) Then, X(t + dt) can be approximated as X(t + dt) = X(t) + dx(t) X(t) + E(dX(t)) + V ar(dx(t)). The variance matrix has a unique square root since it is positive definite. Denoting this square root by V ar(dx(t)) = B dt, the system of SDEs can now be written as : ds = ( β N SI)dt + B 11dW 1t + B 12 dw 2t where W 1 and W 2 are independent Wiener processes. di = ( β N SI γi)dt + B 21dW 1t + B 22 dw 2t (16) 8

9 3.5 Experimental Results and Observations SIS Model Figure 7: Realization of No. of Infected People Figure 8: Equilibrium : Deterministic Case 9

10 Figure 9: Equilibrium : Stochastic Case Figure (7) shows two sample paths of the number of infected members of the population. The parameters in the SIS model (11) are set to β = 1, γ = 0.25, b = 0.25, N = 100 and I(0) = 2. These values also yield R 0 = 2. The blue curve shows the trajectory of I(t) in the deterministic case. Since R 0 > 1, it has been seen that the dynamics for the deterministic case approaches an endemic equilibrium. This is shown in figure (8). However, for the stochastic case in figure (9), it is seen that the dynamics of the number of infected individuals in the population approaches the disease-free equilibrium, as t. The blue curve represents the number of susceptible individuals while the red curve represents the number of infected individuals in the total population of N. It can be said that the stochastic noise acts as a negative feedback to the deterministic dynamics, which is therefore responsible for the number of infected individuals in the population tending asymptotically to the disease free equilibrium, as opposed to the endemic equilibrium in the non-stochastic case. Another parameter of interest is the expected duration of the epidemic, which is the time, T, until which I(T ) = 0. It has been shown above that for the stochastic SIS model, the probability of absorption is one, irrespective of the value of R 0 (because the infected population dynamics always tends to the disease free equilibrium). An absorbing state can be informally described as a point of no return, i.e. once the dynamics hits an absorbing state, it stays there. However, the time to absorption may be short or long depending on the initial conditions and the total size of the population. Let τ(y) be the expected time until absorption starting from an infected population of y (0, N). Then, τ(y) is the solution to the following boundary value problem : (b(y) d(y)) dτ(y) dy + b(y) + d(y) d 2 τ(y) 2 dy 2 = 1 (17) with τ(0) = 0 and dτ dy y=n = 0 and b(y) = (N y)( βy N ) and d(y) = (b + γ)y. While this method provides the desired values of the mean persistence time, it involves the solution of a boundary value problem, which may not always be the best route to adopt. In such a scenario, the average persistence time is more easily found from the transition matrix of the Markov Chain for a birth-death process. The mean persistence time is given by τ p = Q 1 1, where Q is the transition matrix and 1 is a vector of ones, the size of which is equal to the number of states. One can argue that this method is also computationally intensive as it involves the computation of a matrix inverse. However, for the birthdeath process, the transition matrix is sparse (each row has no more than three nonzero elements), which 10

11 means the inverse is easier to compute. Moreover, this matrix has a tridiagonal structure. Efficient recursive algorithms have been developed to compute the inverse of such matrices. On the other hand, the second order differential equation above has to be discretized and the parameter has to be calculated iteratively. The accuracy of the result in this case is heavily dependent on the time step, a problem not faced in the former case SIR Model Figure 10: Realization of No. of Infected People Table 1: Final Size of Epidemic R 0 Det Stoch Figure (10) shows realizations for the number of infected people for the SIR model (12). Here, b = 0, while the other parameters were the same as in the SIS experiment above. The number of infected people was seen to increase monotonically for a certain amount of time, before decreasing monotonically to zero. Like in the earlier case, the blue curve represents the dynamics for the deterministic case while the green and red curves show realizations for the stochastic setting. When b = 0, it is also clear that the number of infected people in the system as t will be zero. The size of the epidemic is defined as the number of cured people at the final time. For an initial infected number I(0) = 1 and total population N = 100 with γ = 1 and b = 0, the table (1) shows the size of the epidemic for various values of R 0 in the deterministic and stochastic cases. The final size of the epidemic in the stochastic case is naturally less than that in the deterministic case for the same value of R 0. However, in both cases, the final size increases with R 0. 11

12 4 The Lotka-Volterra Predator Prey Model ([8]) The Lotka-Volterra model is a set of first order nonlinear differential equations modeling the dynamics of two species that interact with each other. This model has been particularly used to describe environments where one species serves as food for the other (for instance, rabbits and foxes). In the deterministic setting, these equations are given by : The following observations can be made from the above equations : dx 1 (t) = a 11 X 1 (t) a 12 X 1 (t)x 2 (t) (18) dx 2 (t) = a 22 X 2 (t) + a 21 X 2 (t)x 1 (t) (19) 1. X 1 models the dynamics of the population of the prey (rabbits) while X 2 models that of the predator (foxes). 2. The rate of change of population is proportional to its size. 3. The prey finds ample food at all times. 4. The food supply of the predator is entirely dependent on the prey. 5. No environmental factors play a role in this interaction. The change in the population of the prey is given by its own growth less the rate at which it is preyed upon. Similarly, the change in population of the predator is given by the rate at which it feeds on the prey less the rate at which it dies out. The equilibrium points of the system can be found by setting the dynamics in equations (18 and 19) to zero. The origin (0, 0) is naturally one equilibrium point. The second equilibrium point is given by ( a22 a 21, a11 a 12 ). This means that the population will not change if the initial dynamics start at the equilibrium points. In the stochastic setting, environmental noise is added to the dynamics, and the model is represented by : dx 1 (t) = a 11 X 1 (t) a 12 X 1 (t)x 2 (t) + σ 11 X 2 1 (t)dw 1 (t) + σ 12 X 1 (t)x 2 (t)dw 2 (t) (20) dx 2 (t) = a 22 X 2 (t) + a 21 X 2 (t)x 1 (t) + σ 22 X 2 2 (t)dw 2 (t) + σ 21 X 1 (t)x 2 (t)dw 1 (t) (21) 12

13 4.1 Experimental Results and Observations Figure 11: Equilibrium Dynamics of Deterministic L-V Model Figure 12: Deterministic Dynamics of Deterministic L-V Model 13

14 Figure 13: Stochastic Dynamics of Deterministic L-V Model Figure 14: Conserved Quantity : Deterministic Case 14

15 Figure 15: No Conserved Quantity : Stochastic Case For this set of experiments, all parameters in equations (20 and 21) are set to be equal to 1. Figure (11) shows the dynamics of the predator and prey in equilibrium in the deterministic case, i.e. when X 10 = X 20 = 1. The population of each is seen to remain constant with time. Figure (12) shows the population dynamics for the deterministic case when the initial number of predators in the system is 2 (red curve) and the initial number of prey is 4 (blue curve), i.e. X 10 = 4 and X 20 = 2. The dynamics are of an oscillatory nature and they seem to follow each other. In the stochastic framework, with noise added to the system, figure (13) shows that population of the predator eventually falls to 0. Figure (14) plots the number of predators against the number of prey. The closed curve indicates that an energy like quantity is conserved in the system. The fact that this curve is closed also provides an explanation for the oscillatory nature of the dynamics in figure (12). Figure (15) plots the same quantities, but in the presence of environmental noise. In this case, there is no quantity being conserved in that the curve is not closed, and it is seen that the population of the predator goes down to zero. The presence of noise, which prevented the explosion of the population in the two dimensional logistic equation, was also responsible for the extinction of a species in the predator-prey model. 5 The Common Thread In all the models seen so far, the addition of noise was seen to stabilize an otherwise unstable deterministic system. The noise acts as a feedback to the autonomous system, which can then be viewed as a dynamical system being driven by Brownian motion. This section offers some insight into the theory behind some of the results observed in previous sections. To this end, the use of state feedback in the stabilization of linear time invariant systems is presented first, and is subsequently generalized to nonlinear systems driven by Brownian noise. 15

16 5.1 (Full) State Feedback for Deterministic Systems Consider a linear time invariant deterministic dynamical system. The dynamics can be represented as a first order differential equation as : ẋ = Ax x(0) = x 0 (22) The solution to this equation is x(t) = x 0 e At. Here, e At is called the state transition matrix. It is a well known result ([7]) that for the system to be stable, that is for x(t) 0 as t, all the eigenvalues of A must have negative real part. If at least one eigenvalue has a positive real part, then the system (more precisely, the mode corresponding to this eigenvalue) is not stable. Now consider the single input system : ẋ = Ax + Bu x(0) = x 0 (23) where A has at least one eigenvalue with positive real part. In this case, under some assumptions on the matrices A and B (the system should be controllable), the system can be stabilized by an appropriate choice of the input u. In this context, u is called a control. If the states of the system, x, are available for feedback, u can be chosen to be equal to Kx. The modified dynamical system is now ẋ = (A + BK)x, x(0) = x 0. The matrix K can therefore be chosen such that all eigenvalues of (A + BK) have negative real part. This technique is called full state feedback or pole placement. As an aside, it must also be stated that solving for the matrix K in the case when there is more than one input is nontrivial. The solution may not be unique. Choosing the best K will involve the solving of an optimization problem to minimize a cost criterion. 5.2 Stabilization (and Destabilization) by Brownian Noise ([5]) The last two paragraphs dealt with the stabilization of a linear time invariant system using state feedback. This notion is now extended to the more general case of a system with nonlinear dynamics and driven by Brownian noise Lipschitz Functions ([3]) Definition 5.1 A function, f(x, t), f : R d R R p is Lipschitz in x if there exists a constant K > 0 such that f(x, t) f(y, t) p K x y d for every x, y R d. Example 5.2 f(x) = (x 2 + 5), x R is Lipschitz because f is everywhere differentiable and the absolute value of the derivative is upper bounded by 1. Thus, K = 1. Example 5.3 f(x) = x 2, x R is locally Lipschitz, but not globally Lipschitz. Example 5.4 f(x) = x, x [0, 1] is not Lipschitz because the derivative approaches as x tends to Equilibria and Stability ([3]) Consider the dynamical system ẋ = f(x), f : R d R, x(t 0 ) = x 0. Definition 5.5 x e R d is an equilibrium point of the system if f(x e ) = 0. Definition 5.6 x e is stable if for every ɛ > 0, there exists δ = δ(ɛ) > 0 such that x 0 x e δ implies x(t) x e ɛ for all t > t 0. Otherwise, the equilibrium point is unstable. Definition 5.7 x e is convergent if there exists δ 1 > 0 such that x 0 x e δ 1 implies that x(t) x e as t. 16

17 Definition 5.8 x e is asymptotically stable if it is both stable and convergent. Definition 5.9 x e is locally exponentially stable if there exist constants α > 0, λ > 0 such that x(t) x e α x 0 x e e λt for all t > 0, whenever x 0 x e δ. Clearly, exponential stability implies asymptotic stability, but the converse is not true. Let (w 1 (t),..., w m (t)) T be an m dimensional Brownian motion. Let f(x, t), f : R d R + R d be locally Lipschitz and additionally, let f(x, t) K x (24) for all x R d and for all t 0 with K > 0. Consider a stochastically perturbed version of the system ẋ = f(x(t), t), x(0) = x 0 R d : dx(t) = f(x, t)dt + B k x(t)dw k (t) (25) where B k are d d constant matrices. Under the above conditions, equation (25) admits a unique solution x(t; x 0 ). x = 0 being an equilibrium point, implies that f(0, t) = 0, which admits the (trivial) solution x(t) 0. Theorem 5.10 Let equation (24) hold and λ > 0 and ρ 0. Assume : Then, lim sup t B k x 2 λ x 2 (26) x T B k x 2 ρ x 4 (27) 1 t log x(t; x 0) (ρ K 0.5λ) (28) almost surely for any x 0 0. Moreover, if ρ > K 0.5λ, then equation (25) is almost surely exponentially stable. Example 5.11 Let B k = σ k I, where I is the identity matrix and σ k are constants. Equation (25) now becomes: dx(t) = f(x, t)dt + σ k x(t)dw k (t) (29) The conditions of the above theorem can now be written as : B k x 2 = σ 2 x 2 x T B k x 2 = σ 2 x 4 From the above theorem, it can be seen that almost surely lim sup t 1 t log x(t; x 0) (0.5 σ 2 K) The system is almost surely exponentially stable if 0.5 m σ2 > K. 17

18 Example 5.12 In the previous example, if σ k = 0 for 2 k m, the system is almost surely exponentially stable if σ 2 1 > K. Thus, it is evident that the system can be stabilized by a scalar Brownian motion. The following result summarizes the previous statement : Theorem 5.13 Any nonlinear system ẋ(t) = f(x(t), t) can be stabilized by a Brownian motion provided equation (24) is satisfied. Moreover, it can be stabilized by the use of just a scalar Brownian motion. A couple of remarks are in order : Remark 5.14 In a more general setting, a stochastic perturbation can be used to stabilize a stochastic system under conditions similar to those described above. However, such systems are not dealt with in this report. Remark 5.15 Analogous results exist to show that any nonlinear system can be destabilized by Brownian motion, provided d 2. The last remark begets the question about the possibility of destabilizing a stable one dimensional system by Brownian motion. Example 5.16 Consider the one dimensional linear SDE : dx(t) = µx(t) + σ k x(t)dw k (t) with initial condition x(0) = x 0 R. For the unperturbed system to be stable, it is evident that µ < 0. This SDE has the unique solution : which yields : x(t) = x 0 exp{(µ 0.5 σk)t 2 + σ k W k (t)} lim sup t 1 t log x(t; x 0) = µ 0.5 almost surely. Therefore, the unperturbed stable system cannot be destabilized by a linear Brownian motion. 5.3 Theory vs. Numerical Simulation It was mentioned earlier that the logistic equation for dimensions higher than one does not have a closed form solution. While this proves to be a hindrance in performing a detailed analysis at every time step, the asymptotic performance of the numerical algorithms can be analyzed effectively if certain bounds are assumed on the drift and diffusion coefficients of the SDE. It can be clearly seen that in all the models considered in this report, the drift terms (coefficients of dt) are locally Lipschitz. It is also easy to see that the diffusion terms in each case satisfy the conditions in equations (26 and 27). The only thing that remains to be verified is if the coefficients of the dt term satisfy equation (24). In each case, such a K can be easily found as simulations are carried out only for a finite amount of time. This ensures that the systems can be stabilized by Brownian noise. Under these conditions, from Theorem (5.13), it can be seen that a scalar Brownian motion is enough to stabilize every system considered in this report. This also agreed with experimental results where there was only one nonzero Brownian noise term. While it is true that for deterministic models, experimental observations serve as a means to verify theoretical results, it can be confidently said that the reverse is generally true in the stochastic setting. The absence of closed form solutions was a major hindrance in formulating a solid theoretical foundation for the analysis of such models. The convergence of numerical algorithms (observed experimentally), however, has played a huge role in the development of theoretical insights and perspectives on SDE models. σ 2 k 18

19 6 Concluding Remarks and Possible Extensions The modeling of population dynamics by stochastic differential equations was studied. A detailed error analysis was performed for the one dimensional logistic equation by three different methods. In higher dimensions, performing an error analysis was hindered by the absence of a closed form solution to the differential equations. The effect of addition of Brownian noise to the deterministic two dimensional logistic equation, predator-prey and epidemic models was studied in detail. The addition of noise was found to stabilize an initially unstable system. However, this proved to be a double edged sword- in one case, it was seen that the addition of Brownian noise prevented explosion of the population in finite time, while in another, the addition of noise had the effect of making one species in a competitive set up extinct. In the epidemic model, the addition of noise had the effect of reducing the number of infected people in the population to zero asymptotically, while this value asymptotically reached a non zero constant in the deterministic case. Finally, some theoretical perspectives were given into some of the phenomena observed in the numerical simulations. 6.1 Future Work Similar analysis can be carried out for interactions among more than two species. This would lead to a higher dimensional logistic equation. A natural extension would be the modeling of a food chain, where plants grow in abundance uninterrupted, but are consumed by herbivorous species in the food chain. The herbivorous species growth is linked intimately to their interaction with plants. In turn, the herbivores are food for the carnivores, whose growth is proportional to their interaction with the herbivores. The noise in the system could be due to factors like weather conditions and other unforeseen circumstances. All the models considered in this report assumed some form of continuity on the drift and diffusion coefficients. However, in populations, it is not uncommon to observe sudden spurts or dips in numbers of a species. This can occur due to disasters like earthquakes and hurricanes. In such a situation, a jump process must be introduced into the dynamics of the model, in addition to the environmental noise which is modeled by Brownian motion. These jumps can be more accurately modeled by Lévy processes. A Lévy process can be simulated as the sum of a linear Brownian motion and a compound Poisson process. References [1] Linda JS Allen. An introduction to stochastic epidemic models. In Mathematical Epidemiology, pages Springer, [2] Desmond J Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM review, 43(3): , [3] Hassan K Khalil. Nonlinear systems, Third Edition. Prentice hall Upper Saddle River, [4] Peter E Kloeden and Eckhard Platen. Numerical solution of stochastic differential equations, volume 23. Springer Verlag, [5] Xuerong Mao. Stochastic stabilization and destabilization. Systems & control letters, 23(4): , [6] Xuerong Mao, Glenn Marion, and Eric Renshaw. Environmental brownian noise suppresses explosions in population dynamics. Stochastic Processes and Their Applications, 97(1):95 110, [7] Wilson J Rugh. Linear system theory. Prentice-Hall, Inc., [8] Shankar Sastry. Nonlinear systems: analysis, stability, and control, volume 10. Springer New York,