Management of spatial populations Alan Hastings, Environmental Science and Policy, University of California, Davis, USA

Two classes of problems Invasive species control Marine management Key message – use approach appropriate to the particular problem – do not fall into the ‘If I have a hammer, everything looks like a nail’ trap

Problem One Spartina alterniflora Native to eastern US (and Gulf) Invasive in western U.S. 2 sites –S.F. Bay – replacing native –Willapa Bay – invading bare ground

Willapa National Wildlife Refuge

Aerial photos courtesy of Washington State DNR

Spatially-explicit Stochastic Simulation –Consequences of an Allee effect –Compare to analytic model to justify use of latter in designing control strategies Analytical Non Spatial Model –Finding Optimal Control Strategies Models

Low density plants set < 10 X the seed Davis, Taylor, Civille and Strong Journal of Ecology 92 Field Results: Allee effect

Spatially-explicit simulation model 1000m One square km Parameterized from GIS maps (Civille) Field data (Davis, Taylor, Civille, Grevstad) Run for 100 years, time step 1 year Clones have low seed production Meadows have high seed production

Allee Effect Slows Invasion Taylor, Davis, Civille, Grevstad and Hastings. Ecology 2004 Allee Effect No Allee Effect Time (years)

Analytical Non-spatial Model Seedling Area Meadow Area Clone Area Parameters are dependent on density and numbers of individuals. : FECUNDITY OF CLONES : FECUNDITY OF MEADOWS : GROWTH RATE OF CLONES : GROWTH RATE OF MEADOWS : MERGE RATE OF CLONES INTO MEADOWS

Analytical model predicts same dynamics as simulation model Time (years) Area Occupied (km 2 ) No Allee effect Allee effect Non-spatial analytical model Simulation model

Control of Spartina

2003 Control

Control questions How much Spartina needs to be removed every year to eradicate invasion within 10 years Is it better to prioritize removal of fast growing but low seed producing clones or is it better to prioritize removal of slow-growing but high seed producing meadows?

Control Strategy T t < MAX = Total area removed in year t 0  X t  1 = fraction of T t that was meadows 0  (1- X t )  1 = fraction of T t that was clones

Control Objectives 1.Eradicate invasion in one square km region within 10 years 2.Minimize Cost X Risk of colonizing other sites Total area removed in 10 years Total seed production during 10 years

Minimum Removal needed to Eradicate within 10 years Equivalent of 15-20% of initial invasion has to be removed annually Initial Area of Invasion Minimum annual removal Taylor and Hastings. Journal of Applied Ecology 2004

Optimal Control Strategies High Budget Meadows First 3,600 m 2 /year5,000 m 2 /year Low Budget Clones First

Switch Control Strategies High Budget Clones First 3,600 m 2 /year5,000 m 2 /year Low Budget Meadow First

Summary Low BudgetHigh Budget Minimize Cost Only Clones First Minimize Risk Only Clones FirstMeadows First Minimize Cost and Risk Clones FirstMeadows First No Allee Effect Clones First

Linear control model Density independent Three classes – seedlings,juveniles and adults –Express model in terms of area occupied If the model were nonlinear this would become a dynamic programming problem – –Difficult numerical problem – cannot really get a solution –So, can we simplify in this case? (Hastings, Hall and Taylor, TPB in press)

Population = size without control – contribution of removed

What classes should be removed? One year ahead? –The class that contributes the most area (normalized by ‘cost’) should be removed first “Infinitely” far ahead? –The class that has the highest reproductive values (normalized by ‘cost’) should be removed first Therefore do intermediate case, finite time horizon, which becomes a linear programming problem (from previous slide)

Population size as a function of time and the annual budget allocated to control, when the objective is to minimize the population within 10 years subject to budget constraint.

The fraction of each stage class (green for isolates, red for meadows) removed by control in each year under the optimal control strategy.

Initial conclusions Optimal approach is time dependent –May be much more effective Cost of waiting –Overall cost of control can be much less when started earlier Since a LP problem solution is always at a vertex – focus on a single class unless budget large enough to remove an entire class, then add one more class

‘Easy’ extensions Dependence on habitat Spatial extent Dependence on tidal height

Damage (Hall and Hastings, submitted)

Conclusions Allee effect slows down invasion considerably Best control strategy is to remove clones first if budget is low or if minimizing for cost only If minimizing for risk and budget is high, removing meadows first is best strategy Meadow first strategy is risky especially if budgets for future years are unpredictable.

Where’s the data? Willapa Bay –Analysis of aerial photographs SF Bay –Remote sensing data

Approach in Willapa Bay Initial steps –(orthorectify, etc.) With aid of GIS software, identify clones –By hand Match up successive years

Invasive Spartina in SF Bay

Approach in SF Bay Low resolution, high bandwidth data Identify components by ‘spectral signature’ –Ground truthing –Choose number of components to identify Mud Water Spartina Other vegetation (Rosso, P. H., Ustin, S. L. & Hastings, A. (2005) International Journal of Remote Sensing 26: 5169 – 5191)

Picture is an Aviris image (pixel size, 17x17 m approx.) of Coyote Creek area marsh, in the southern tip of San Francisco Bay. Image is from August Colors indicate the percentage of each component, Spartina (red), Salicornia (green) and water (blue), present at each pixel as determined by a spectral unmixing approach. The unmixing was done on the basis of eight endmembers (reference spectra). Five plant species, water and open mud.

Comparison of approaches Willapa –High resolution, low bandwidth –High accuracy –Labor intensive –Data expensive –Works well with invasion into bare mud SF Bay –Low resolution, high bandwidth –Lower accuracy –After difficult initial steps, easier to implement –Can handle multiple types

Problem 2: Management of fisheries Persistence (Maximum sustainable yield – not today) Present economic value

There are many issues that may affect the design of marine protected areas Habitat requirements Habitat heterogeneity Species interactions Economic issues Dispersal Fishing pressures Ocean currents Human considerations

There are different ways to incorporate these issues Habitat requirements Habitat heterogeneity Species interactions Economic issues Dispersal Fishing pressures Ocean currents Human considerations Simulation based models Analytic models for general principles

Analytic models have advantages in data poor situations Easier to look at a wide range of assumptions (parameters), and therefore provide a range of alternatives Best way to identify critical data needs and to connect with data General principles can guide the use of simulation models Key is knowing what to leave out

General modeling approaches Integrodifference equations Numbers at x = SUM OVER LOCATIONS y of production at y times probability of moving from y to x Production at y Probability of moving from y to x

Marine Reserves and Persistence Assume that settlement probability depends only on distance from point of release

Reserve Width and Spacing Two nondimensional parameter combinations Fraction in reserves, width in mean dispersal units

Persistence Botsford, Gaines, and Hastings, Ecol Letters 2001 Focus on recruitment Want larvae settling in reserves to be at least 35% of settlement if no harvest Assume that all recruitment provided by adults in reserves

Fraction of Natural settlement as a function of fraction in reserves and reserve width as calculated by two estimates to show when system persists, general results used in MLPA process in California Required for persistence Below persistence

The Bahamas

Model structure-density independent (for persistence)- Hastings and Botsford (PNAS 2006) p 21 Production a i, Survival of settling larvae b i Persistence determined by matrix C with c ij =a i p ij b j p 12 p 23 p 31 p 32 p connectance

Single species, multi-patch persistence can depend on network contributions c ij =b j p ij a i (‘exchange’ terms) Assume no single patch persists (Use Perron-Frobenius and Q=C-I to change criteria from eigenvalue < 1 to real part <0, then use results from M-Matrices leading to..) Network persistence criteria on next slide

Network persistence for 2 and 3 patch systems – have results for arbitrary number of patches Shortfall of local patches, or residency time Contribution back to patches after 2 years Return to original patch after 3 years 2 patch subnetworks q ij =c ij =b j p ij a i 4 patch condition similar but adds new features and has 26 terms

Model structure-density independent (for persistence)- p 12 Production a i, Survival of settling larvae b i p 21 p 23 p 13 p 32 p connectance

Connectivity Patch 2 persists on its own Patch 1 persists on its own Patch 3 persists on its own

Need connectivity and life history parameters to get persistence Simple model shows that merely preserving habitat types where a species is now found not enough to ensure persistence –Self-connectivity would depend on size of habitats preserved – Model shows that species can be found currently in locations that do not contribute to persistence –Model shows how to calculate persistence in general –Model shows that patches that either just retain larvae, or just disperse larvae well do not contribute to network persistence – must do both

Economic analysis of fishing in 2 patch model Sanchirico, Malvadkar, Hastings, and Wilen, Ecol. Appl. (2006) Look at two patches –allow movement of fish as both juveniles and adults –Consider a specific ecological model, include heterogeneity in fishing costs and ecological dynamics –Consider spatial fishing effort that produces maximum present value

Consider all possible combinations of harvest rates (efforts) in two patches

The optimal harvesting solution can lead to the closure of a fishing area

Simple models have greatly increased our understanding of the role of marine protected areas Data requirements for determining persistence are clear –Network effects may be important –Even patches with low production may be important –Need patches that both collect and produce Reserves can play a key economic role in fisheries –With MSY analysis, can identify when yields are equivalent (Hastings and Botsford, 1999) –With maximizing present value, optimal solution can be closure of part of the habitat, i.e., a reserve

Simple models can play a key role Decisions made in response to societal demands require solutions that –Satisfy multiple stakeholders –May need to be done before complete studies Examples –MLPA in California –Bahamas –Invasive species control