MWF 11-12:10, 169 Baskin Engineering
This year, the focus of my graduate course in mathematical biology will be the quantitative methods for fish population dynamics and fishery management (these methods apply to management of any renewable resource, although fisheries science and ecology have different terms for the same things).
I have a number of goals: to set you up for courses in Bayesian analysis so that you can develop models with meaningful dynamics, to give background that will make more advanced applied mathematics courses (e.g. in control theory, pdes or odes) more meaningful, and to get you ready to read the literature in a sophisticated way and do to research.
The approximate topical outline is
Production models, elementary bioeconomics, the acronym soup
Age, growth and maturity: von Bertalanffy growth and its extensions Life history invariants
Yield per recruit, Spawning Per Recruit, Spawning Stock Biomass Per Recruit. Spawning Potential Ratio
Stock and recruitment
Age structured models: Forward dynamics
Formalisms for incorporating uncertainty
Evolution of steepness
Salmon life history and fishery management
Fisherman behavior via stochastic dynamic programming
Spatial dynamics and the theory of marine reserves
Investment and advanced bioeconomics
Prerequisites for this course are a working knowledge of calculus, knowledge of ordinary differential equations and probability theory comparable to a first undergraduate course in each. In addition, you will need to be able to program a computer. The language is unimportant – the problems can be done in any language from EXCEL(used by Ray Hilborn for calculations he did in The Ecological Detective) or BASIC (required by Andre Punt for his quantitative methods course at UW) through R, MATLAB, or C++. However, I will not teach programming.
Homework will be given during lecture and homework from one week will be due at the start of class on the subsequent Monday (Wednesday if Monday is a holiday). My policy about late homework is simple: it will not be accepted. I will not have office hours Monday morning (i.e. before the homework is due). I will grade on a scale of 0, 1, 2, 3 or 4 points and letter grades will be assigned (<65% C or worse; 65-85%: B, > 85%:A). Homework will count for 70% of the grade.
The final examination will consist of three or four common problems (which will be indicated during the lectures) and a modest project. The final will be given out on 1 March and is due by 5 pm on 16 March. It will count for 20% of the grade. The remaining 10% of the grade will be determined by in-class participation..
Literature: Books
This is a list of books that is good to know about, some of which are worth dreaming about owning (even as a graduate student). I believe that an excellent personal library is an essential tool for mathematical biology; books such as these are our supplies.
Beverton, R.J.H. and S.J. Holt. 1954 (1993 reprint) On the Dynamics of Exploited Fish Populations. Chapman and Hall (put this on your wish list. Had they had today's computing power, Beverton and Holt could have done everything).
In 1994, shortly before his death, Ray Beverton gave a series of lectures in Woods Hole, MA. These are really wonderful and public domain. So you can get it now.
Bulmer, M. 1994. Theoretical Evolutionary Ecology. Sinauer Associates
Charlesworth, B. 1994. Evolution in Age-structured Populations. 2nd edition. Cambridge University Press (put this on your wish list)
Clark, C.W. 1985. Bioeconomic Modelling and Fisheries Management. Wiley Interscience(put this on your wish list)
Clark, C.W. 1990. Mathematical Bioeconomics. The Optimal Management of Renewable Resources. Wiley Interscience (put this on your wish list)
Colin is writing a new book "Worldwide Crisis in Fisheries. The Ise and Misuse of Models" and has graciously allowed us to us the penultimate draft. You can get Chapter 1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6 now.
Clark, C.W.. and M. Mangel. 2000. Dynamic State Variable Models in Ecology. Methods and Applications. Oxford University Press (well, it is on my wish list that you have it)
Cushing, J.M. 1998. An Introduction Structured Population Dynamics. SIAM (this is a good, but mathematically tough, book).
Edelstein-Keshet, L. 1988. Mathematical Models in Biology. Random House (I think that this may now be reprinted by SIAM).
Elliott, J.M. 1994. Quantitative Ecology and the Brown Trout. Oxford University Press (put this on your wish list)
Farlow,S.J. 1993. Partial Differential Equations for Scientists and Engineers. Dover (just buy this; it is only 16 bucks)
Fisher, R.A. 1930 (1999). The Genetical Theory of Natural Selection. Oxford University Press (put this on your wish list)
Groot, C. and L. Margolis. 1991. Pacific Salmon Life Histories. UBC Press, Vancouver
Gurney, W.S.C. and R.M. Nisbet. 1998. Ecological Dynamics. Oxford University Press (put this on your wish list)
Hart, P.J.B. and J.D. Reynolds. Handbook of Fish Biology and Fisheries. Volume 1 (Fish Biology), Volume 2 (Fisheries). Blackwell Science.
Hilborn, R. and M. Mangel. 1997. The Ecological Detective. Confronting models with data. Princeton University Press. (also on my wish list)
Hilborn, R. and C. J. Walters. 1992. Quantitative fisheries stock assessment : choice, dynamics, and uncertainty. Chapman and Hall.
Hoppensteadt, F.C. 1982. Mathematical Methods of Population Biology. Cambridge University Press.
Houston, A.I. and J.M. McNamara. 1999. Models of Adaptive Behavior. Oxford University Press
Jennings, S., Kaiser, M.J. and J.D. Reynolds. 2001. Marine Fisheries Ecology. Blackwell Science
Kot, M. 2001. Elements of Mathematical Ecology. Cambridge University Press. (put this on your wish list).
Mangel, M. and C.W. Clark. 1988. Dynamic Modeling in Behavioral Ecology. Princeton University Press (also on my wish list)
McCall, A.D. 1990. Dynamic Geography of Marine Fish Populations. University of Washinton Press (add to your wish list)
McEvoy, A.F. 1986. The Fisherman's Problem. Ecology and Law in the California Fisheries.1850-1980. Cambridge University Press (put this on your wish list)
Murdoch, W.W., Briggs, C.J. and R.M. Nisbet. 2003. Consumer-Resource Dynamics. Princeton University Press (put this on your wish list)
Murrary, J.D. 2002. Mathematical Biology. I: An Introduction. Springer Verlag (put this on your wish list)
Murrary, J.D. 2003. Mathematical Biology. II: Spatial Models and Biomedical Applications. Springer Verlag
Nisbet, R.M. and W.S.C. Gurney. 1982. Modelling Fluctuating Populations. Wiley Interscience
Quinn, T.J.H. and R.B. Deriso. 2000. Quantitative Fish Dynamics. Oxford University Press
T.D. Smith. 1994. Scaling Fisheries. The science of measuring the effects of fishing, 1855-1955. Cambridge University Press(put this on your wish list)
Walters, C.J. and S.J.D. Martell. 2004. Fisheries Ecology and Management. Princeton University Press, Princeton, NJ
Literature: Journal Articles.
I have assembled a modest list of papers that I think are worth knowing about. I will refer to many, but not all, of them in lectures. Almost all of these are available through the California Digital Library and since the papers are copyrighted and this is an open web site, I am not posting them here. If you are registered for the class and need a paper not on the CDL, let me know and I can send it to you.
I will surely add to this list as the term goes on.
Production models, elementary bioeconomics, the acronym soup
Frank, K. T., and W. C. Leggett. 1994. Fisheries Ecology in the Context of Ecological and Evolutionary Theory. Annual Review of Ecology and Systematics 25:401-422.
Getz, W.M. 1998. An introspection on the art of modeling in population ecology. BioScience 48:540-552 (if somebody gets a pdf of this, I would appreciate it).
Goulder, L. 2002. An eye on the future. Nature 419:673-674
Harley, S.J., Myers, R.A. and A. Dunn. 2001. Is catch-per-unit-effort proportional to abundance? Canadian Journal of Fisheries and Aquatic Sciences 58:1760-1772
Hutchings, J.A. and J.D. Reynolds. 2004. Marine fish population collapses: consequences for recovery and extinction risk. BioScience 54:297-309
Maunder, M. 2003. Is it time to discard the Schaefer model from the stock assessment scientist's toolbox? Fisheries Research 61:145-149
Quinn, T.J. 2003. Ruminations on the development and future of population dyanmics models in fisheries. Natural Resource Modeling 16:341-391
Reed, M. C. 2004. Why is mathematical biology so hard? Notices of the AMS 51:338-342.
Runge, M.C. and F.A. Johnson. 2002. The importance of funcitonal form in optimal control solutions of problems in population dynamics. Ecology 83:1357-1371
Schnute, J. 2003. Designing fishery models: A personal adventure. Natural Resource Modeling 16:393-413.
Age, growth and maturity: von Bertalanffy growth and its extensions Life history invariants
Beverton, R. H., and S. J. Holt. 1959. A review of the lifespans and mortality rates of fish n nature, and their relaion to growth and other physiological characteristics. Pages 142-174 in G. E. W. Wolstenholme and M. O'Connor, editors. CIBA Foundation Colloquia on Ageing. CIBA Foundation, London.
Beverton, R. J. H. 1987. Longevity in fish: some ecological and evolutionary considerations. Pages 161-185 in A. D. a. T. Woodhead, K.H., editor. Evolution of Longevity in Animals. Plenum, New York.
Bokma, F. 2004. Evidence against universal metabolic allometry. Functional Ecology 18:184-187.
Beverton, R. J. H. 1992. Patterns of reproductive strategy parameters in some marine teleost fishes. Journal of Fish Biology 41 (Supplement B):137-160.
Essington, T.E. Kitchell, J.F. and C.J. Walters. 2001. The von Bertalanffy growth function, bioenergetics, and the consumption rate of fish. Canadian Journal of Fisheries and Aquatic Sciences 58:2129-2138
Frisk, M.G., Miller, T.J. and M.J. Fogarty. 2001. Estimatikon and analysis of biological parameters in elasmobranch fishes: a comparative life history study. Canadian Journal of Fisheries and Aquatric Sciences 58: 969-981
Gunderson, D.R. 1997. Trade-off between reproductive effort and adult survival in oviparous and viviparous fishes. Canadian Journal of Fisheries and Aquatic Sciences 54:990-998
Hoenig, J. M. 1983. Empirical use of longevity data to estimate mortality rates. Fishery Bulletin 81:898-903.
Hutchings, J.A. 1993. Adaptive life histories effected by age-specific survival and growth rate. Ecology 74:673-684
Hutchings, J.A. 1999. Influence of growth and survival costs of reproduction on Atlantic cod, Gadus morhua, population growth rate. Canadian Journal of Fisheries and Aquatic Sciences 56:1612-1623
Jennings, S., J. D. Reynolds, and S. C. Mills. 1998. Life history correlates of responses to fisheries exploitation. Proceedings of the Royal Society of London B 265:333-339.
King, J.R. and G.A. McFArlane. 2003. Marine fish life history strategies: applications to fishery managem,ent. Fisheries Management and Ecology 10:249-264
Lorenzen, K. 2000. Allometry of natural mortality as a basis for assessing optimal release size in fish-stocking programmes. Canadian Journal of Fisheries and Aquatic Sciences 57:2374-2381
Mangel, M. 1996. Life history invariants, age at maturity and the ferox trout. Evolutionary Ecology 10:249-263 (you can get this at my web site, but not through the CDL).
McGurk. M. 1986. Some remarks on "Model of monthly marine growth and natural mortality for Babline Lake sockeye salmon (Oncorhynchus nerka) " by Furnell and Brett. Canadian Journal of Fisheries and Aquatic Sciences 43:2535-2537
Stamps, J.A., Mangel, M. and J.A. Phillips. 1998. A new look at relationships between size at maturity and asymptotic size. American Naturalist 152:470-479
von Bertalanffy, L. 1957. Quantitative laws in metabolism and growth. Quarterly Review of Biology 32:217-231.
West, G. B., J. Brown, H., and B. J. Enquist. 2001. A general model for ontogenetic growth. Nature 413:628-631.
Yield per recruit, Spawning Per Recruit, Spawning Stock Biomass Per Recruit. Spawning Potential Ratio
Brodziak, J. 2002. In search of optimal harvest rates for west coast groundfish. North American Journal of Fisheies Management 22:258-271.
Clark, W. G. 1991. Groundfish exploitation rates based on life history parameters. Canadian Journal of Fisheries and Aquatic Sciences 48:734-750.
Clark, W. G. 2002. F35% revisited ten years later. North American Journal of Fisheies Management 22:251-257.
Stock and Recruitment
Barrowman, N.J. and Myers, R.A. 2000. Still more spawner recruitment curves: the hockey stick and its generalizations. Can.J. Fish. Aquat. Sci. 57: 665&endash;6763
Bjorkstedt, E.P. 2000. Stock-recruitment relationships for life cycles that exhbiti concurrent density dependence. Canadian Journal of Fisheries and Aquatic Sciences 57:459-467
Brodziak, J. K. T., W. J. Overholtz, and P. J. Rago. 2002. Reply: Does spawning stock affect recruitment of New England groundfish? Interpreting spawning stock and recruitment daa in New England 11 groundfish. Can. J. Fish. Aquat. Sci. 59:193-195.
Mace, P.M. and I.J. Doonan. 1988. A generalized bioeconomic simulation model for fish population dynamics. N.Z. Fish. Assess. Res. Doc. 88/4. 17 (I regret that I do not have this as a pdf; if somebody gets it that way, I should appreciate receiving it)
McAllister, M. K., and G. P. Kirkwood. 1998. Bayesian stock assessment: a review and example application using the logistic model. ICES J. Mar. Sci. 55:1031-1060.
Needle, C.L. 2002. Recruitment models: diagnosis and prognosis. Reviews in Fish Biology and Fisheries. 11: 95-111.
Rochet, M.J. 2000. Does the concept of spawner per recruit make sense? ICES J. Mar. Sci. 57: 1160-1174
Schnute, J.T. and A.R. Kronlund. 1996. A management oriented approach to stock recruitment analysis. Canadian Journal of Fisheries and Aquatic Sciences 53:1281-1293.
Age structured models: Forward dynamics
Clark, W.G.1999. Effects of an erroneous natural mortality rate on a simple age-structured stock assesment. Canadian Journal of Fisheries and Aquatic Sciences 56:1721-1731
Higgins, K., A. Hastings, and L. W. Botsford. 1997. Density dependence and age structure: Nonlinear dynamics and population behavior. American Naturalist 149:247-269.
Huang, X-C. 1990. An age-dependent population model and its operator. Physica D 41:356-370 (This is a serious math paper)
Gurney, W.S.C., Nisbet, R.M. and J.H. Lawton. 1983. The systematic formulation of tractable single-species population models incorporating age structure. Journal of Animal Ecology 52:479-495
Gurney, W.S.C., Speris, D.C., Wood, S.N., Clarke, E.D. and M.R. Heath. 2001. Simulating spatially and physiologically structured populations. Journal of Animal Ecology 70:881-894
Schnute, J.T. and L.J. Richards. 1998. Analytical models for fishery reference points. Canadian Journal of Fisheries and Aquatic Sciences 55:515-528
Tucker, S.L. and S.O. Zimmerman. 1988. A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM Journal on Applied Mathematics 48:549-591 (This is a serious math paper)
Formalisms for incorporating uncertainty
Calder, C., Lavine, M. , Müller, P. and J.S. Clark. 2003. In corporating multiple sources of stochasticity into dynamic population models. Ecology 84:1395-1402
Clark, J.S. 2003. Uncertainty an variability in demography and population growth: a hierarchial approach. Ecology 84:1370-1381
Cortes, E. 12002. Incorporating uncertainty into demographic modeling: Application to shark poulations and their conservation. Conservation Biology 16:1048-1062
DeValpine, P. and A. Hastings. 2002. Fitting population models incorporating process noise and observation error. Ecological Monographs 72:57-76
Foley, P. 1994. Predicting extintion times from environmental stochasticity and carrying capacity. Conservation Biology 8:124-137
Hoper, K.R., Rosenheim, J.A., Prout, T. and S.J. Oppenheim. 2003. Within-generatiion bet-hedging: a seductive explanation? Oikos 101:219-222
Leslie, P.H. and J.C. Gower. 1958. The properties of a stochastic model for two competing species. Biometrika 45:316-330
Lewontin, R.C. and D. Cohen., 1969. On populastion growthj in a randomly varying environment. PNAS 62:1056-1060
Mangel, M. 1994. Barrier transitions driven by fluctuations, with applicationsto ecology and evolution. Theoretical Population Biology 45:16-40
Maunder, M.N. 2001. A general frameowrk for integrting the standardization of catch per unit effort into stock assessment models. Canadian Journal of Fisheries and Aquatic Sciences 58:795-803
McAllister, M.K., Pikitch, E.K. and E.A. Babcok. 2001. Using demographic methods to construct Bayesian priors for the intrinsic rate of increase in the SChaefer models and implications for stock rebuilding. Canadian Journal of Fisheries and Aquatic Sciences 58:1871-1890
McNam ara, J.M. 1997. Optimal life histories for structured populations in fluctuating environments. Theoretical Population Biology 51:94-108
Millar, R.B. 2002. Reference priors for Bayesian fisheries models. Canadian Journal of Fisheries and Aquatic Sciences 59:1492-1502
Rees, M., Mangel, M., Turnbull, L., Sheppard, A., and D. Briese. 2000. The effects of heterogeneity on dispersal and colonization in plants. Pg 237-265 in The Ecological Consequences of Environmental Heterogeneity (MJ Hutchings, EA John and AJA Stewart, editors), Blackwell Science, Oxford UK
Evolution of Steepness
Ashley, M.V., Willson, M.F., Pergams, O.R.W., O'Dowd, D.J., Gende. S.M. and J.S. Brown. 2003. Evolutionarily enlightened management. Biological Conservation 111:115-123
Dorn, M.W. 2002. Advice on west coast rockfish harvest rate from Bayesian meta-analysis of stock-recruitment relationships. North American Journal of Fisheries Management 22:280-300.
Kirby, D.S., Fiksen Ø. and P.J.B. Hart. 2000. A dynamic optimisation model for the behaviour of tunas at ocean fronts. Fisheries Oceanography 9:328-342
McNamara, J.M. 1997. Optimal life histories for structured populations in fluctuating environments. Theoretical Population Biology 51:94-108
McNamara, J.M., Houston, A.I. and E.J. Collins. 2001. Optimality models in behavioral biology. SIAM Review 43:413-466
Millar, R.B. 2002.Reference priors for Bayesian fisheries models.Canadian Journal of Fisheries and Aquatic Sciences 59:1492-1502.
Myers, R.A., N.J. Barrowman, R. Hilborn, and D.G. Kehler. 2002. Inferring Bayesian priors with limited direct data: Applications to risk analysis. North American Journal of Fisheries Management 22:351-364.
Rose, K.A., Cowan, J.H., Jr., Winemiller, K.O., Myers, R.A. and R. Hilborn. 2001. Compensatory density dependence in fish populations: importance, controversy, understanding and prognosis. Fish and Fisheries. 2:293-327
Salmon life history and fishery management
Bradford, M.K. 1995. Comparative review of Pacific salmon survival rates. Canadian Journal of Fisheries and Aquatic Sciences 52:1327-1338
Hill, M. F., L. W. Botsford, and A. Hastings. 2003. The effects of spawning age distribution on salmon persistence in fluctuating environments. Journal of Animal Ecology 72:736-744.
Schnute, J.T. and A.R. Kronlund. 2002. Estimating salmon stock-recruitment relationships from catch and escapement data. Canadian Journal of Fisheries and Aquatic Sciences 59:433-449
Mangel, M. 1994. Climate change and salmonid life history variation. Deep Sea Research, II (Topical Studies in Oceanography) 41:75-106
Thorpe, J.E. 1994. An alternative view of smolting in salmonids. Aquaculture 121:105-113
Thorpe, J.E. 1994. Salmonid fishes and the estuarine environment. Estuaries 17:76-93
Thorpe, J.E., Mangel, M., Metcalfe, N.B. and F.A. Huntingford. Modelling.1998.The proximate basis of salmonid life-history variation, with application to Atlantic salmon, Salmo salar L. Evolutionary Ecology 12:581-600
Young, T.P. 1981. A general model of comparative fecundity for semelparous and iteroparous life histories. American Naturalist 118:27-36
Fisherman behavior via stochastic dynamic programming
Gillis, D. M., E. K. Pikitch, and R. M. Peterman. 1995. Dynamic Discarding Decisions: Foraging Theory for High-Grading in a Trawl Fishery. Behavioral Ecology 6:146-154.
Gillis, D. M., and R. M. Peterman. 1998. Implications of interference among fishing vessels and the ideal free distribution to the interpretation of CPUE. Canadian Journal of Fisheries and Aquatic Sciences 55:37-46.
Gillis, D. M. 1999. Behavioral inferences from regulatory observer data: catch rate variation in the Scotian Shelf silver hake (merluccius bilinearis) fishery. Can. J. Fish. Aquat. Sci. 56:288-296.
Gillis, D. M. 2003. Ideal free distributions in fleet dynamics: a behavioral perspective on vessel movement in fisheries analysis. Canadian Journal of Zoology 81:177-187.
Mangel, M. and D. Ludwig. 1992. Definition and evaluation of the fitness of behavioral and developmental programs. Annual Review of Ecology and Systematics. 23:507-536
Spatial dynamics and the theory of marine reserves
Botsford, L., D. Kaplan, and A. Hastings. 2004. Sustainability and yield in marine reserve policy. American Fisheries Society Symposium 42:75-86.
Botsford, L. W., A. Hastings, and S. D. Gaines. 2001a. Dependence of sustainability on the configuration of marine reserves and larval dispersal distance. Ecology Letters 4:144-150.
Guichard, F., Levin, S.A., Hastings, A. and D. Siegel. Toward a dynamic metacommunity approach to marine reserve theory. BioScience 54:1003-1011
Hastings, A., and L. W. Botsford. 1999a. Equivalence in yield from marine reserves and traditional fisheries management. Science 284:1537-1538
Holland, D.S. 2003. Integrating spatial management measures into traditional fishery management systems: The case of the Georges Bank multispecies groundfish fishery. ICES Journal of Marine Science 60:915-929
Lauck, T., Clark, C.W., Mangel, M. and G.R. Munro. 1998. Implementing the precautionary principle in fisheries management through marine reserves. Ecological Applications 8(1) Supplement: S72-78.
Lockwood, D. R., A. Hastings, and L. W. Botsford. 2002a. The effects of dispersal patterns on marine reserves: Does the tail wag the dog? Theoretical Population Biology 61:297-309.
Lundberg, P. and N. Jonzen. 1999. Spatial population dynamics andthe design of marine reserves. Ecology Letters 2:129-134
Lundquist, C. J., and L. W. Botsford. 2004. Model projections of the fishery implications of the Allee effect in broadcast spawners. Ecological Applications 14:929-941.
Mangel, M. 1998. No-take areas for sustainability of harvested species and a conservation invariant for marine reserves. Ecology Letters 1:87-90
Mangel, M.2000. On the fraction of habitat allocated to marine reserves. Ecology Letters 3:15-22
Mangel, M. 2000. Irreducible uncertainties, sustainable fisheries and marine reserves.Evolutionary Ecology Research 2:547-55
Mangel, M. 2000.Trade-offs between fish habitat and fishing mortality and the role of reserves. Bulletin of Marine Science 66:663-674
Skellam, J.G. 1951. Random dispersal in theoretical populations. Biometrika 38:196-218
van Kirk, R.W. and M.A. Lewis. 1999. Integrodifference models for persistence in fragmented habitats. Bulletin of Mathematical Biology 59:107-137
Investment and advanced bioeconomics
MORE TO BE ADDED ONCE I KNOW THE INTERESTS AND ABILITIES OF THE CLASS
Ricker's Definition of MSY
MAXIMUM SUSTAINABLE YIELD (MSY OR YS): The largest average catch or yield that can continuously be taken from a stock under existing environmental conditions. (For species with fluctuating recruitment, the maximum might be obtained by taking fewer fish in some years than in others.) Also called: maximum equilibrium catch (MEC); maximum sustained yield; sustainable catch.
Ricker, W.E. 1975. Computation and Interpretation of Biological Statistics of Fish Populations.. Bulletin 191, Fisheries Research Board of Canada
Larkin's (1977) Epitaph for MSY
M.S.Y.
1930s-1970s
Here lies the concept, MSY.
It advocated yields too high,
And didn't spell out how to slice the pie.
We bury it with best of wishes.
Especially on behalf of fishes.
We don't know yet what will take its
place.
But hope it's as good for the human race.
Things We Learned From Production Models (class composite)
•Endogenous bioeconomic cycles exist
•MSY is marginally stable and should be a constraint, not a target
• Harvest equations vary widely; one must know assumptions and think critically
• Knowing population size is important for setting harvest constraints
•Discounting value inherently promotes overharvesting
• Different levels of effort can result in the same yield
•Economic optimization can drive populations to extinction
• There exist different perspectives and definitions on what is "optimal"
• Bioeconomic equilibrium population size is independent of biology
• Any model in which population sizes can become negative or probabilities can fall outside of [0,1] must be wrong
• We can learn a lot from the Schaefer model but should not use it for management
• Production depends on the characteristics of density dependence
• CPUE proportional to N? Not all the time!
•There is a "bonus" in harvesting down from K to a stable population size
• This "bonus" will lead to overcapacity
Life must be lived forwards, but can only be understood backwards
Soren Kierkegaard