Download - Catalyst

Spatial models
(meta-population models)
Readings
• Hilborn R et al. (2004) When can marine reserves
improve fisheries management? Ocean and Coastal
Management 47:197-205
• Hilborn R et al. (2006) Integrating marine protected
areas with catch regulation. CJFAS 63:642-649
Overview
• Why worry about space?
• General metapopulation model
• One-dimensional model of marine protected areas
Why worry about space?
Dutch beam trawl fleet
England
Netherlands
Rijnsdorp AD et al. (1998) Micro-scale distribution of beam trawl effort in the southern North Sea between 1993 and 1996 in relation to
the trawling frequency of the sea bed and the impact on benthic organisms. ICES Journal of Marine Science 55:403-419
UK fisheries
Beam trawlers
Dredgers
Netters
Otter trawlers
Potters
All combined
Jennings S & Lee J (2012) Defining fishing grounds with vessel monitoring system data. ICES Journal of Marine Science 69:51-63
Ideal free distribution
Abundance
(variable)
CPUE
(constant)
Effort
(variable)
Swain DP & EJ Wade (2003) Spatial distribution of catch and effort in a fishery for snow crab
(Chionoectes opilio): tests of predictions of the ideal free distribution. CJFAS 60:897-909
MPA
MPA
MPA
Marine
protected
area
(MPA)
MPA
Murawski SA et al. (2005) Effort distribution and catch patterns adjacent to temperate MPAs. ICES Journal of Marine Science 62:1150-1167
Reason for “fishing the line”
Dollars per hour
CV of dollars
Hours trawled
Murawski SA et al. (2005) Effort distribution and catch patterns adjacent to temperate MPAs. ICES Journal of Marine Science 62:1150-1167
Metapopulation models
A metapopulation is a series of discrete
populations that are isolated but have limited
exchange.
Generalized metapopulation model
Emigration: movement out
of area i into area j
Numbers in area i in time t
Ni ,t 1  f  Ni ,t , ei ,t    Eij   E ji
j
Environmental
conditions in area
i in time t
j
Immigration: movement
from area j into area i
Important details
• Population dynamics within an area
• Models of dispersal
• Environmental models, process error, catastrophic
events
Boundaries?
• Absorbing boundary
http://www.google.com/pacman/
– Disappear/die on hitting the boundary
– E.g. another country, advection into open ocean, natural
range bounds
• Reflective boundary
– Remain in the cell next to the boundary
– E.g. mountain range, river barrier
• Pac-man (circular coastline)
– Reappear on the opposite side
– E.g. circumpolar, islands
One-dimensional logistic-growth
model with harvesting
Numbers in area i in time t
Density
dependence
Harvest
Logistic model
 Ni ,t 
Ni ,t 1  Ni ,t  Ni ,t ri  1 

Ki 

Exploitation rate
 ui ,t Ni ,t
Immigration
 m [1  ui 1,t ]Ni 1,t  [1  ui 1,t ]Ni 1,t 
Emigration
 2m[1  ui ,t ]Ni ,t
Cell on the left
Only survivors of harvest will move
Movement rate the
same in all cells
Cell on the right
Harvest, then movement
16 spatial models animation.r
Diffusion scenario
• 21 areas, 50 time steps, migration rate 0.2, r = 0.2, K
= 1000, reflective boundary
• Diffusion scenario: starting population = K in center
cell, 0 in all other cells, exploitation rate 0
16 spatial models animation.r
Diffusion, no harvesting, N11 = K
Abundance
(black = first year, light gray = last year)
Cell number
16 spatial models animation.r
Marine protected area scenario
• As before: 21 areas, 50 time steps, migration rate
0.2, r = 0.2, K = 1000, reflective boundary
• MPA scenario: starting population = K in all cells,
exploitation rate 0 in center 5 cells (numbers 9–13),
exploitation rate 0.2 in all other cells
16 spatial models animation.r
MPA: no harvest in center cells
Abundance
(black = first year, light gray = last year)
Cell number
16 spatial models animation.r
Do protected areas increase yields?
• 51 areas, migration rate m = 0.2, r = 0.2, K = 1000,
start population = K in all areas
• Run with u = 0, 0.01, 0.02, …, 0.9
• After a large number of time steps (1000) the model
is at equilibrium, and yield in the final time step is
the equilibrium yield for each value of u
• No MPA: all areas have harvest rate = u
• MPA: 5 middle areas have zero harvest rate, other
have harvest rate = u
16 MPA yield.r
16 MPA yield.r
Yield in each year
Time series of catches
Initially the no MPA scenario has higher
catches; after year 14 with high u = 0.25
the MPA has higher catches; with no MPA,
the population can go extinct
No MPA
Almost at
equilibrium
yield
u = 0.1
MPA
u = 0.25
Year
16 MPA yield.r
Do protected areas increase yield?
Higher yield
without MPA
u = r = 0.2
uMSY = r/2 = 0.1
Equilibrium yield
No MPA
Yield maximized
at higher harvest
rate with MPA
MPA
At u > r, MPA
halts collapse
(insurance
policy)
Harvest rate (u)
16 MPA yield.r
Break-even point
uMSY with MPA
uMSY without MPA
Equilibrium yield
Zoomed in
No MPA
MPA
Harvest rate (u)
16 MPA yield.r
Bigger MPA (close 25 cells)
(previously, close 5 of 51 cells)
Equilibrium yield
No MPA
MPA
Closing more
areas reduces the
maximum yield
more
Maximum yield
reduced from
2550 to 1377 in
this simulation
Harvest rate (u)
16 MPA yield.r
For the same yield what is u?
Equilibrium yield
No MPA
MPA
MPA: u = 0.10
produces MSY
No MPA: two
values of u result
in equivalent
yield: u = 0.032
and u = 0.168
Harvest rate (u)
16 MPA yield.r
For the same yield, what is biomass?
Abundance
No MPA
u = 0.032
MPA
u = 0.1
No MPA
u = 0.168
Cell number
16 MPA yield.r
For the same yield, what is CPUE?
(assume effort is proportional to harvest rate u)
No MPA
u = 0.032, yield = 1370, CPUE = 1370/0.032 = 42800
MPA
u = 0.1, yield = 1370, CPUE = 13700
No MPA
u = 0.168, yield = 1370, CPUE = 8200
16 MPA yield.r
Lessons
• Closing areas reduces the maximum yield
• The more areas closed, the lower the maximum yield
• Closed areas provide insurance against high fishing
pressure (bad management, lack of enforcement)
• For every level of yield with an MPA there is an
equivalent yield without an MPA which has:
–
–
–
–
–
higher biomass outside the MPA
lower biomass inside the MPA
lower harvest rate where fishing occurs
higher CPUE and hence greater profits
no completely protected areas
Old target
New target
Percent of maximum
Tradeoffs of fishing
Exploitation rate
Worm B et al. (2009) Rebuilding global fisheries. Science 325:578-585
Lessons
•
•
•
•
Spatial scale and pattern matters
Simple movement models
Marine protected areas: insurance vs. yield
Trade-offs between catch, profit, and biodiversity