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Advection control in parabolic PDE systems for competitive populations

  • Author Bio: E-mail address: rkokum@gmail.com; E-mail address: phan@math.utk.edu; E-mail address: lenhart@math.utk.edu
  • * Corresponding author: lenhart@math.utk.edu

    * Corresponding author: lenhart@math.utk.edu
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  • This paper investigates the response of two competing species to a given resource using optimal control techniques. We explore the choices of directed movement through controlling the advective coefficients in a system of parabolic partial differential equations. The objective is to maximize the abundance represented by a weighted combination of the two populations while minimizing the 'risk' costs caused by the movements.

    Mathematics Subject Classification: Primary:49K20, 92B05;Secondary:35K57.

    Citation:

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  • Figure 1.  Different resource functions $m(x)$

    Figure 2.  Different initial conditions: (2a) Smaller initial population at middle, (2b) Larger initial population at middle, (2c) Two smaller initial populations overlapping in the middle

    Figure 3.  One population only: Population dynamics and optimal control for $u$ population with the resource function $m=x/5$ as in Figure 1a; (3a) Optimal control $h_1$ with respect to the IC in Figure 2a, (3b) Population distribution of $u$ with respect to the IC in Figure 2a, (3c) Optimal control $h_1$ with respect to the IC in Figure 2b, (3d) Population distribution of $u$ with respect to the IC in Figure 2b

    Figure 4.  One population only: Population dynamics and optimal control for $u$ population with the resource function $m=sin(\pi x/5)$ as in Figure 1b; (4a) Optimal control $h_1$ with respect to the IC in Figure 2a, (4b) Population distribution of $u$ with respect to the IC in Figure 2a, (4c) Optimal control $h_1$ with respect to the IC in Figure 2b, (4d) Population distribution of $u$ with respect to the IC in Figure 2b

    Figure 5.  One population only: Population dynamics and optimal control with a smaller IC as in Figure 2a and larger resources $m=6x/5$ as in Figure 1c; (5a) Optimal control $h_1$, (5b) Population distribution of $u$

    Figure 6.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2a and the resource function $m=x/5$ as given in Figure 1a; (6a) Optimal control $h_1$, (6b) Population distribution of $u$

    Figure 7.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2a and the resource function $m=sin(\pi x/5)$ as given in Figure 1b; (7a) Optimal control $h_1$, (7b) Population distribution of $u$

    Figure 8.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2c and the resource function $m=x/5$ as given in Figure 1a; (8a) Optimal control $h_1$, (8b) Population distribution of $u$, (8c) Optimal control $h_2$, (8d) Population distribution of $v$

    Figure 9.  Two populations with competition: Population dynamics and optimal control with a smaller IC as in Figure 2c and the resource function $m=sin(\pi x/5)$ as given in Figure 1b; (9a) Optimal control $h_1$, (9b) Population distribution of $u$, (9c) Optimal control $h_2$, (9d) Population distribution of $v$

    Figure 10.  Two populations with different competition rates: Population dynamics and optimal control with a smaller IC as in Figure 2a and the resource function $m=sin(\pi x/5)$ as given in Figure 1b and $b_1 = 4$, $b_2 = 0.5$; (10a) Optimal control $h_1$, (10b) Population distribution of $u$, (10c) Optimal control $h_2$, (10d) Population distribution of $v$

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