Design of fishing exclusion zones Phil Broadbridge, Colin Please, - PowerPoint PPT Presentation

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Design of fishing exclusion zones Phil Broadbridge, Colin Please, Ashleigh Hutchinson, Bothwell Maregere, Roy Gusinow, Michael M c Phail, Ebrahim Fredericks January 23, 2019 1 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Outline Problem statement 1 Single species model 2 Effect of geometry 3 Improvements 4 Further work 5 Conclusions 6 2 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Problem statement The goal is to: Ensure the survival of endangered fish by achieving balance between fishing, birth, and movement rates. Compare the effectiveness of different geometries of exclusion zones. Improve on the current mathematical model by modifying the source term. Finding parameter values using known data. 3 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Single species model Reaction-diffusion equation: θ t = −∇ . q + S ( θ , x ) . (1) Using Fick’s law we get θ t = ∇ . [ D ( θ ) ∇ θ ] + f ( x ) R ( θ ) . (2) Possible modifications: Consider a time-dependent domain. Using the Reynolds transport theorem we can show that equation (1) still holds with q = − D ( θ ) ∇ θ + θ u where u is the velocity of the domain. Previous models for circular geometries use f ( r ) = 1 and R ( θ ) = s θ ( 1 − θ / m ) . We will consider a Gaussian distribution. 4 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Comparison of different geometries y = b r = a Exclusion zone y = 0 x = 0 x = a We first linearise the governing equation for small θ and small |∇ θ | which gives θ t = D ( 0 ) ∇ 2 θ + s θ . (3) 5 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Comparison of different geometries Now consider a rectangular domain with length a and width b . For now let f ( x , y ) = 1. Use separation of variables to get a basis of solutions. We find that θ = A exp [ A lm t ] sin ( xl π / a ) sin ( ym π / b ) , (4) where � m 2 π 2 + l 2 π 2 � A lm = − D 0 + s . (5) b 2 a 2 For A 11 > 0, the population will increase. This gives a result in terms of the hyperbolic rms length 1 � D ( 0 ) / s . > 2 π (6) � 1/ ( 2 b 2 ) + 1/ ( 2 a 2 ) 6 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Comparison of different geometries Rearranging allows us to write this expression as a ratio of area to diagonal ab � √ a 2 + b 2 > π 2 D ( 0 ) / s . (7) For a square we have � D ( 0 ) / s a > 2 π (8) Reduction of 2-D model to 1-D model gives � D ( 0 ) / s . a > π (9) For a circular geometry � a > λ 1 D ( 0 ) / s , J 0 ( λ 1 ) = 0. (10) 7 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Infinite series of zones Danger Danger Exclusion 8 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Dimensional model Symmetry Symmetry θ t = D 2 ¯ ¯ θ xx + B ( 1 − α ) ¯ ¯ θ x = 0 θ t = D 1 θ xx + B θ θ θ x = 0 x 1 x 2 0 θ = ¯ θ D 1 θ x = D 2 ¯ θ x 9 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Dimensionless model Employ the scaling t = x 2 x = x 1 x ′ , 1 t ′ D 1 . Symmetry Symmetry θ t = D ¯ ¯ θ xx + ˆ B ( 1 − α ) ¯ ¯ θ x = 0 θ t = θ xx + ˆ θ x = 0 B θ θ 0 1 x ˆ θ = ¯ θ θ x = D ¯ θ x 10 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Dimensionless model Employ the scaling t = x 2 x = x 1 x ′ , 1 t ′ D 1 . Symmetry Symmetry θ t = D ¯ ¯ θ xx + ˆ B ( 1 − α ) ¯ ¯ θ x = 0 θ t = θ xx + ˆ θ x = 0 B θ θ B = x 2 x = x 2 1 B ˆ ˆ x 1 D 1 D = D 2 D 1 0 1 x ˆ θ = ¯ θ θ x = D ¯ θ x 11 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Solution in zone I Variable separable solution for the population density function θ = X ( x ) T ( t ) furnishes the general solution in I λ 2 t � � T = T ( 0 ) exp (11) Two cases B − λ 2 > 0 ˆ Case 1 : X ( x ) = A 1 cos ( ω 1 x ) , (12) B − λ 2 < 0 ˆ Case 2 : X ( x ) = A 1 cosh ( ω 1 x ) , (13) where ω 2 1 = Abs [( ˆ B − λ 2 )] . 12 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Solution in zone II Variable separable solution for the population density function θ = X ( x ) T ( t ) unearths the general solution in II λ 2 t � � T = T ( 0 ) exp (14) Two cases, but we are really interested in the case where α > 1. � � Case A : X ( x ) = A 1 cosh ( λ x ) − tanh ( λ x 2 ) sinh ( λ x ) (15) where 2 = ˆ B [( 1 − α ) − λ 2 ] − λ (16) 13 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Interface conditions The dimensionless interface conditions at x = x 2 are θ = θ , θ x = D θ x . (17) For Case 1 and Case A this leads to � ˆ � ˆ �� ˆ � � � B − λ 2 coth B − ˆ ( ˆ x − 1 ) B α − λ 2 B − λ 2 tan = 1. � ˆ B − ˆ B α − λ 2 D (18) For λ = 0: � √ ˆ � ˆ � � � B − ˆ ( ˆ x − 1 ) B α B coth tan D √ = 1. (19) 1 − α 14 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Condition for marginal stability The hyper-surface in ( ˆ x ) space defined by B , α , D , ˆ � √ ˆ � � � � ˆ ( ˆ x − 1 ) B ( 1 − α ) coth tan B D √ = 1, (20) 1 − α separates region of different stability. Can use (20) to evaluate conservation strategies. 15 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Conservation strategy Survival Excluded fraction Large zones Small zones x 1 Figure: Fraction of river that must be excluded. Sample parameters: D = 1, α = 5, and B = 1 16 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Stability analysis for modified reactive diffusion model Environmental heterogeneous factor is a parabolic curve. Model 1 − ǫ x 2 � � θ t = D θ xx + B θ (21) Variable separable solution θ = e At Q ( x ) , where � 1 � 2 a + 1 4, 1 2, 1 Q ( x ) = e − 1 4 x 2 M 2 x 2 (22) and a = 1 A − B 2 ( B ǫ ) − 1 2 1 2 D ( 0 ) 17 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Linear stability analysis θ t = D ( 0 ) θ xx + s θ f ( x ) , θ = P ( t ) Q ( x ) , A = P ′ ( t ) = D ( 0 ) Q ′′ ( x ) + s ( 1 − ǫ x 2 ) p Q D ( 0 ) − ǫ sx 2 � s − A � Q ′′ + Q = 0, D ( 0 ) � 1 4 . � D ( 0 ) x = x 1 ℓ , ℓ = √ ǫ s 2 � a + 1 � Q ′′ − 4 x 2 Q = 0, � D ( 0 ) � 1 2 A − s a = 1 D ( 0 ) . 2 ǫ s 18 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions First approximation linear stability analysis We consider the even solution � 1 � 2 a + 1 4, 1 2, 1 Q ( x ) = l − 2 e − 1 4 x 2 M 2 x 2 , First zero approximation (Abramowitz Stegun) occurs when � 2 π 2 � 1 + 1 4 − 3 x 2 4 = , 1 − 2 a − 1 π 2 x 2 ¯ = − 4 a . Stability cross-over � D ( 0 ) � 1 2 2 x 1 > π , s which concurs with findings above. Next correction by the Newton-Raphson method is less than 1% ( f = 1 at bddry). 19 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Finding parameter values using data We have 6000 data points for fish movement. Fit the probability density function: P = α ( π D 1 t ) − 1/2 exp [ − x 2 /4 D 1 t ]+ ( 1 − α )( π D 2 t ) − 1/2 exp [ − x 2 /4 D 2 t ] . (23) So we have two sub-populations, namely, home-bodies and travellers which was considered in the previous model. 20 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Statistical confirmation Data maintained on Oceanographic Research Institute’s Cooperative Fish Tagging Project. Need to confirm model with paper by Bruce et al 2016. x i 2 The diffusivity coefficient D = 2 π t i will be furnished by random walk analysis. 21 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Numerical simulations Lie point symmetries of the non-linear equations. Perturbation methods. Numerical solutions of the full non-linear equations. Question the applicability of Fick’s law. Consider a moving boundary. 22 / 23

Problem statement Single species model Effect of geometry Improvements Further work Conclusions Conclusions The square is the best geometry! A simple 1-D model provides a simple framework for assessing conservation strategies. Including the parabolic adjustment gives no further information at the lowest order. ”So long and thanks for all the fish!” 23 / 23