Advection control in parabolic PDE systems for competitive populations

Kokum R. De Silva; Tuoc V. Phan; Suzanne Lenhart

doi:10.3934/dcdsb.2017052

Article Contents

2017, Volume 22, Issue 3: 1049-1072. Doi: 10.3934/dcdsb.2017052

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Advection control in parabolic PDE systems for competitive populations

1.
Department of Mathematics, University of Peradeniya, Peradeniya, KY 20400, Sri Lanka
2.
Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USA

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

Received: May 2016

Revised: December 2016

Published: September 2017

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Abstract

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.

Keywords:

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

Citation:

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

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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

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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

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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

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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 $u$

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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 $u$

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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 $u$

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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 $v$

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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 $v$

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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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