Investigating advection control in parabolic PDE systems for competitive populations Kokum De Silva Department of Mathematics University of Peradeniya, Sri Lanka Tuoc Phan and Suzanne Lenhart Department of Mathematics University of Tennessee, Knoxville, USA May 18, 2016
OUTLINE Background Formulation of optimal control problem Optimality Conditions Numerical results
Introduction The reaction of a population for resources is an important concern in ecology. With inhomogeneous resources population may tend to move along the spatial gradient of the resource function depending on the initial conditions. What about when there is more than one population and there is a competition among them to survive?? http://383326907.r.worldcdn.net/wp-content/uploads/2012/04/question-mark-man.jpg?5c86f3
The Goal The Question: In an area with two competing populations, if a population could choose the direction for advection movement, how would such a choice be made to maximize its total population? The Goal: To investigate this question with optimal control of PDE
Motivation and previous work Previous work: Concentrating on Movement Given a fixed amount of resources, how does the species react to the habitat to be beneficial? Movement: Random Diffusion and Directed Advection. Belgacem-Cosner and Cantrell-Cosner-Lou studied the effects of the advection along an environmental resource gradient u t [D u αu m(x)] = λu[m(x) u], Ω (0, ) with zero flux boundary condition. m(x) represents the intrinsic growth rate and measures the availability of the resources. beneficial means the persistence of the population or the existence of a unique globally attracting steady state.
Motivation and previous work Previous work: Population Dynamics Model If a species could choose the direction for advection movement, how would such a choice be made to maximize its total population? REFERENCE: Finotti, Lenhart, Phan, EECT 2013 Ω R n is a bounded smooth domain, Q T = Ω [0, T ] and S T = Ω [0, T ] for some fixed T > 0. Model with u(x, t), population density u t [µ u u h] = u[m f (x, t, u)], Q T, µ u ν u h ν = 0, S T, u(, 0) = u 0 0, Ω. h : QT R n is the advection direction. m = m(x, t) in L (Q T ) measures the availability of resources. f : Q T R R is non-negative and has some smoothness and growth conditions. µ > 0 fixed (diffusion coefficient) and u 0 L (Ω) is smooth.
Motivation and previous work Previous work: Problem Formulation Seek the advection term h(x, t) control that maximizes the total population while minimizing the cost due to movement. Find h U such that J( h ) = sup J( h), where J( h) = U Q T [u(x, t) B h(x, t) 2 ]dxdt. U = { h L 2 ((0, T ), L 2 (Ω) n ) : h k M, k = 1, 2,, n}. B cost coefficient due to the population moving along h. Limited numerical results with small initial conditions (with little variation) show optimal controls following the gradient of m.
Problem formulation In an area with two competing populations, if the populations could choose the direction for advection movement, how would such a choice be made to maximize its total population? Let u(x, t), v(x, t) be the population densities of two competing species in a spacial domain Ω in d - dimensional space R d, d N. Assume Ω with smooth boundary Ω and For a given fixed time 0 < T < let Q = Ω (0, T ) and. S = Ω (0, T )
The system of PDE s describing the dynamics: u t d 1 u ( h 1u) = u[m a 1u] b 1uv in Q v t d 2 v ( h 2v) = v[m a 2v] b 2uv in Q u d 1 η + u h 1 η = 0 on S (1) v d 2 η + v h 2 η = 0 on S u(x, 0) = u 0(x) 0 for x Ω v(x, 0) = v 0(x) 0 for x Ω
The optimal control problem formulation: Control Set U = {( h 1, h 2) ((L (Q)) d, (L (Q)) d ) : (h 1) i M 1, (h 2) i M 2, i = 1,..., d} and (u, v) = (u( h 1, h 2), v( h 1, h 2)) be the solution of (1) for the corresponding ( h 1, h 2). Then, for A, B, C, D 0 we find ( h 1, h 2) U such that J(( h 1, h 2) ) = sup J( h 1, h 2) ( h 1, h 2 ) U where, J( h 1, h 2) is the objective functional given by, J( h 1, h [ 2) = Au + Bv C h 1 2 D h 2 2] dxdt (2) subject to the PDE system (1). Q We maximize a weighted combination of the two populations while minimizing the cost due to the movements of the populations. The cost is due to the risk of movements.
Existence and positivity of the state solution and the existence of an optimal control Solution space u, v L 2 ((0, T ), H 1 (Ω)) H (Q) with u t, v t L 2 ((0, T ), (H 1 (Ω)) ) Theorem Given m L (Q), u 0, v 0 non-negative, L (Q) bounded and in H 1 (Ω). Then, for each ( h 1, h 2) U, there is a unique positive weak solution (u, v) = (u( h 1, h 2), v( h 1, h 2)) of the state system (1). Theorem There exists an optimal control ( h 1, h 2) maximizing the functional J( h 1, h 2) over U. i.e. J( h 1, h 2) = sup ( h 1, h 2 ) U J( h 1, h 2).
Derivation of the optimality system To derive the necessary conditions, we need to differentiate the map We differentiate the ( h 1, h 2) J( h 1, h 2) map and the ( h 1, h 2) (u, v)( h 1, h 2) map with respect to the controls h 1 and h 2. The derivatives of this ( h 1, h 2) (u, v)( h 1, h 2) map are the sensitivity functions ψ 1 and ψ 2
The sensitivity PDE s For a given control ( h 1, h 2) U, consider another control ( h 1, h 2) ɛ = ( h 1 + ɛ l 1, h 2 + ɛ l 2) s.t. ( h 1, h 2) + ɛ( l 1, l 2) U for all sufficiently small ɛ > 0 with ( l 1, l 2) ((L (Ω)) d, (L (Ω)) d ) and We can obtain u ɛ = u( h 1 + ɛ l 1, h 2 + ɛ l 2) and v ɛ = v( h 1 + ɛ l 1, h 2 + ɛ l 2) u ɛ u, v ɛ v, u ɛ t u t, v ɛ t v t u ɛ u ɛ ψ 1 and vɛ v ɛ ψ 2 in L 2 ((0, T), H 1 (Ω))
Sensitivity PDE s (ψ 1) t d 1 ψ 1 ( h 1ψ 1 + l 1u) = mψ 1 2a 1uψ 1 b 1vψ 1 b 1uψ 2 in Q (ψ 2) t d 2 ψ 2 ( h 2ψ 2 + l 2v) = mψ 2 2a 2vψ 2 b 2vψ 1 b 2uψ 2 in Q d 1 ψ 1 η + ψ1 h 1 η = u l 1 η on S (3) d 2 ψ 2 η + ψ2 h 2 η = v l 2 η on S ψ 1(x, 0) = 0 for x Ω ψ 2(x, 0) = 0 for x Ω.
The adjoint functions: To characterize the optimal control, we will use the sensitivity function together with adjoint function to differentiate ( h 1, h 2) J( h 1, h 2) map. The adjoint functions show how the states u, v affect the goal So, we find the formal adjoint of the operator in the sensitivity PDE s.t. Left hand sides of weak forms of sensitivity equations with adjoints as test functions = Left hand sides of weak forms of the adjoint equations with sensitivities as test functions The non-homogeneous terms on the RHS of the adjoint equations are obtained as, (Integrand ofj) (Au+Bv C h A 1 2 D h 2 2 ) u u = = B (Integrand ofj) v (Au+Bv C h 1 2 D h 2 2 ) v
The adjoint PDE s (λ 1) t d 1 λ 1 + h 1 λ 1 mλ 1 + 2a 1uλ 1 + b 1vλ 1 + b 2vλ 2 = A in Q (λ 2) t d 2 λ 2 + h 2 λ 2 mλ 2 + 2a 2vλ 2 + b 1uλ 1 + b 2uλ 2 = B in Q λ 1 η λ 2 η = 0 on S (4) = 0 on S λ 1(x, T ) = 0 for x Ω λ 2(x, T ) = 0 for x Ω.
Characterizing the optimal control h Suppose there exist an optimal control ( h 1, h 2) U with the corresponding states u, v and compute the directional derivative of the function J( h 1, h 2) with respect to ( h 1, h 2) in the direction ( l 1, l 2) at u, v. Since, J( h 1, h 2) is the maximum value for the J( h 1, h 2) we have J(( h 0 lim 1, h 2) + ɛ( l 1, l 2)) J(( h 1, h 2) ) ɛ 0 + ɛ = (Aψ 1 + Bψ 2 2Ch 1 l1 2Dh 2 l2) dxdt Q
Characterizing the optimal control h Hence, for any ( l 1, l 2) we have 0 ( l 1, l 2) (u λ 1 + 2Ch 1, v λ2 + 2Dh 2 ) dxdt Q Using cases of (h 1, h 2) i and the sign of variation (l 1, l 2) i, ( ( )) (h 1) i = min M 1, max u(λ1)x i 2C, M1 ( ( )) (h 2) i = min M 2, max v(λ2)x i 2D, M2 (5) Theorem For sufficiently small T, the solution to the optimality system (1), (4) and (5) is unique.
The numerical results We consider the one dimensional problem corresponds to problem (2): u t d 1u xx (h 1u) x = u[m a 1u] b 1uv on (0, L) (0, T ) v t d 2v xx (h 2v) x = v[m a 2v] b 2uv on (0, L) (0, T ) d 1u x ν + uh 1 ν = 0 for x = 0 or L and t (0, T ) d 2v x ν + uh 1 ν = 0 for x = 0 or L and t (0, T ) u(x, 0) = u 0(x) 0 for x (0, L) v(x, 0) = v 0(x) 0 for x (0, L) Hence from (5) we have h 1 = min ( ( M 1, max u(λ1)x ( ( h2 = min M 2, max 2C )), M1 )). v(λ2)x 2D, M2
We used a finite difference scheme to solve the PDE problem and the advection term was modeled using the up wind method. We used the forward backward sweep method to solve the optimal control problem.
Parameter values: a 1 = 1 a 2 = 1 A = 1 B = 1 C = 0.5 D = 0.5 Diffusion Coefficients d 1 = 0.2 d 2 = 0.2 Control limits maximum h i = 4 minimum h i = 4 Domain: spatial length: 5 units and time length: 2 With our notation for the sign of advection terms, h i being negative (positive) represents movement to right (left).
Different resource functions (a) x/5 (b) sin(πx/5) (c) 6x/5 Figure 1: Different resource functions m(x)
Different initial conditions (a) (b) (c) Figure 2: Different initial conditions 2(a) Smaller initial populations at middle (both same) 2(b) Larger initial populations at middle (both same) 2(c) Two smaller initial populations overlapping in the middle
Without competition: One population only, effects of variation in IC (a) (b) (c) (d) Figure 3: OC for u with m = x/5 and lower IC at top and higher IC at bottom. With higher IC, the OC not driven only by gradient of m
Two populations with competition and same small ICs (a) Optimal control h 1 (b) Population distribution of u Figure 4: Populations and optimal controls with same small ICs and the resource function m = sin(πx/5) with effect of competition on OC
Two populations with competition and different small ICs (a) Optimal control h 1 (b) Population distribution of u (c) Optimal control h 2 (d) Population distribution of v Figure 5: Population dynamics and optimal control with different small ICs and m = sin(πx/5), Different OCs due to ICs and competition
Two populations with different competition rates: b 1 = 4, b 2 = 0.5 (a) Optimal control h 1 (b) Population distribution of u (c) Optimal control h 2 (d) Population distribution of v Figure 6: Population dynamics and optimal control with same small ICs and m = sin(πx/5), Effect of different competition rates
Conclusions With numerical simulations for one population only, we were able to show the population does not always choose the advection direction to move toward increasing resources. When the initial condition has a sufficiently high population with some variation, the movement may be chosen to move to level the population, instead of moving toward increasing resources. In the systems case, the level of the competition coefficients can also influence the choice of movement direction. Advective directions depend on the initial conditions, diffusion effects, competition rates, and resources. Currently working on PDE generalizations including other types of interactions besides competition.
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Without competition: One population only (a) (b) (c) (d) Figure 7: OC for u with m = sin(πx/5) and higher IC in lower plots
Two populations with competition (a) Optimal control h 1 (b) Population distribution of u (c) Optimal control h 2 (d) Population distribution of v Figure 8: Population dynamics and optimal control with a smaller IC as in Figure 2(c) and the resource function m = x/5 as given in Figure 1(a)