The second modeling component of the F3 project to be presented is a spatially explicit fish population dynamic model that allows us to explore the consequences of stochastic dispersal and heterogeneous fishing effort. This model simulates the changes of nearshore populations using an integraldifference approach, or
A x

A n x

H n x

M (A x n x

H ) x
 
(A n x '

H )F L K x ' x ' n x x '
R dx ' x
(1) where n
A = x adult abundance at location x in year n (# / km), rate (# / km), M n x

F = fecundity of adults at x'
H n  x natural mortality for adults(fraction of adults/year), dispersal kernel from location x’ to x in year n (1 / km), is the harvest n
K = x-x' larval location x’ (# larval releases per fecund adult), through settlement), R = post-settlement recruitment of settled larvae to adults at x location x and the integration is over all sites x’ (fraction). For simplicity, the terms for fecundity, larval survival and post-settlement recruitment are multiplied together and are described using a single constant to quantify fish productivity, P o
(# recruits per spawning adult). Density dependence is described using the Ricker equation and can be acting on post-settlement recruitment based on adult densities or larval densities, on fecundity based on adult densities or on adult mortality. There are important differences that arise in the dynamics of a population in how density dependence is applied on the system which are especially important when stochastic dispersal is considered. We are in the process of drafting a manuscript on this particular issue now. Stochastic settlement is modeled assuming by selecting a few draws from a probability distribution of dispersal which is a function of PLD and ocean current conditions
(Siegel et al. 2003). We refer to this as the “spiky” model. The number of draws is a function of the release duration and character as well as the PLD of the organism.
We used circulation models that to add realistic spatial structure to the larval release and settlement. Figure 1A shows simulations from the circulation model over several spawning seasons. We added the source-destination relationships from the circulation model to the population model, referred to as the “packet” model (see Figure 1B). Now, larvae within a set of adjacent sources are re leased together to for “packets” that are advected by the currents and settle together, across adjacent settlement locations. The “packet” model functions

similarly to the “spiky” model but adds spatial correlation to the larval release and settlement sites.

Figure 1 . Connectivity matrices for larval dispersal examining the role of spawning season (each season is 90 days). (A) Connectivity matrices obtained from the simulations of ROMS model. (B) Prediction by a simple advectiondiffusion model. (C) Prediction by the settlement-pulse model. For more than 1 season, this represents the average connectivity over multiple spawning seasons.

Figure 2: Modeled population density and recruitment with and without stochastic settlement for a nearshore fish stock. (a) and (b) spatial distribution for the 50 th generation (after statistical steady state is achieved). The time space history larval settlement (# per year) when statistical steady state occurs (generations 20 to 50) for the
“spiky” model (c) and the “packet” model (d). Model parameters are: M = 0.05, Po =
1100, post-settlement density dependence, no harvest, absorbing boundaries (at +/- 250 km –  x = 5 km), Kernel parameters – PLD = 50 days. Stochastic settlement is modeled using 60 draws each with a 5% probability of success.

Figure 3.
The time space history of adult population for two similar and competing species statistical steady state occurs (generations 300 to 500) for the “packet” model.
Model parameters are: M = 0.09, P
A
is 10% higher than P
B
, post-settlement density dependence which depends on the sum of the settlement of both species, no harvest, absorbing boundaries (at +/- 250 km
–  x = 5 km), Dispersal parameters
– PLD = 30 days , size of settlement “packet” is 50 km.

Harvest is also applied in this system through the implementation of a fishing policy. Polices implemented to date include both spatial and non-spatial policies and can be set knowing a final escapement level, the total allowable catch or a prescribed fishing mortality rate.
The temporal order of processes follows the life cycle introduced previously,

Fished individuals are removed from system, H n x
, based upon a prescribed fishing policy.

Settlers are released into the plankton as a function of adult fecundity at a spawning location and the surviving fish densities (

F (A x ' n x '

H ) x '
). This can be a density dependent process (as expected for sea urchin).

A fraction of these spawned larvae survive (based upon the larval survival probability, L n ) to settle at a location determined by the dispersal kernel,
K n x-x'

A fraction of the settlers recruit to adult stages (density dependence is often important at this stage of the life cycle for many groundfish).

The adult population is reduced by natural mortality ( M n x
) which again may be density dependent.

New recruits are added to remaining adult population in order to generate the census available to the next time step.
This model is written in Matlab and is freely available via the Flow, Fish and
Fishing webpage ( www.icess.ucsb.edu/~satoshi/f3 ). A full description of the modeling system is available there along with a webpage where you can try out the model. We are now at the point of performing science with this model. We have mentioned several activities going on with our fish stock / harvest model and are drafting manuscripts. These papers are on a wide variety of problems.
For example, we find that even in a homogeneous physical environment, the statistics of larval dispersal over the course of a single season are spatially heterogeneous and tempor ally intermittent, for both the “spiky” and “packet” versions of the model. This shows that stochastic larval dispersal will have an important role on the intergeneration changes in fish stocks and will provide difficulties in informing the fishery management process. With stochastic dispersal, harvesting at maximum sustainable yield maximizes the spatial heterogeneity in fish stocks. This is not intuitive but arises due to the type and strength of the density dependence in stock dynamics. If the primary source of density dependence is post-settlement, then marine protected areas can increase fisheries profits, and there is a wide range of nearly equivalent solutions in terms of both total area and reserve configuration. In addition to these types of single species models, we have started modeling the interactions between two similar species that share resources. With stochastic dispersal, the model predicts coexistence, while diffusive dispersal causes one species to out compete the other to extinction. A long-lived adult life history and stochastic

dispersal is enough to promote coexistence between the species, as long they have temporally and/or spatially uncorrelated settlement. This result occurs, regardless of the starting abundance of each species, with both “spiky” and
“packet” dispersal patterns as the less fecund species has an increase in growth rate when rare. We also developing a single species version of the model that includes age structure as well as age/size-dependent harvest. This model is designed to investigate how adding more detailed biological information about the fish species (e.g. increasing fecundity with age) and more complex forms of harvest impact the future productivity of the species, given temporally and spatially variable dispersal. Manuscripts describing all of these issues are underway at this time.