Geometric Population Growth in Discrete Time DN = Nt+1 - Nt = R R = Total Population Growth Rate

Geometric Population Growth: J-shaped Curve of Nt versus Time (t) Population size, Nt , accelerates with time in a J-shaped Curve.

Geometric Population Growth: Plot Rt = l = (Nt+1 – Nt) against Nt. Population Growth Rate, R, is proportional to Population size or Density, Nt. Rt Nt

Geometric or Exponential Growth: Plot of ‘r’ against Nt. Population Growth Rate per capita, r, is constant as population size or density, Nt, changes. r Nt

Exponential Growth: 3 Equivalent Looks Population size accelerates with time: J-shaped Curve. Nt Time Total Population grow rate is proportion to population size or density. R Nt Per capita growth rate, r, is constant as population size or density increases. r Nt

Logistic Population Growth in Discrete Time DN = Nt+1 - Nt = R R = Total Population Growth Rate

Logistic Population Growth: Plot Nt+1 against t: S-shaped curve K Nt K/2 t Logistic Growth: S-shaped Curve

Logistic Population Growth: Plot Rt = l = (Nt+1 – Nt) against Nt. Total Population Growth Rate, R, changes with Population size or Density, Nt. R increases, peaks, and then decreases. K Steepest part of this curve is where N is changing the fastest. Nt K/2 t Logistic Growth: S-shaped Curve

Logistic Population Growth: Plot Rt = l = (Nt+1 – Nt) against Nt. Total Population Growth Rate, R, changes with Population size or Density, Nt. R increases, peaks, and then decreases. K Nt Rt K/2 K/2 K t Nt Logistic Growth: S-shaped Curve

Logistic Population Growth: Plot Rt = l = (Nt+1 – Nt) against Nt Maximum occurs at (K/2), half the carrying capacity. Maximum value of (Nt+1 – Nt) Rt K/2 K Nt

Logistic Population Growth: Plot of ‘r’ against Nt. Population Growth Rate per capita, r, declines as population size or density, Nt, changes. r = rmax([K – Nt]/K) Minimum N, maximum r r rmax Maximum N, minimum r K Nt

Logistic Growth: 3 Equivalent Looks Population size accelerates and then decelerates with time: S-shaped Curve. Nt Time Total Population grow rate changes with population size or density. R Nt Per capita growth rate, r, is declines as population size or density increases. r Nt

Logistic Population Growth: Imagine that you are managing a Natural fishery and you want to harvest the most fish you can. However, the fish you do not eat are going to be the parents of the next generation. If you catch too many fish this year, there will be fewer next year. If you catch too few fish this year, you will not sell as many as you could have sold. How many fish should you catch? How many fish should you leave in the ocean?

MSY: Maximum Sustainable Yield

MSY The maximum sustainable yield or MSY is the largest catch that can be removed from a population of fish every generation for an indefinite number of generations. With logistic population growth, the MSY equals (K/2), half the carrying capacity of the species. This also is the point with the maximum value of R, the total population growth rate.

Logistic Population Growth: Plot Rt = l = (Nt+1 – Nt) against Nt Maximum occurs at (K/2), half the carrying capacity. Maximum value of (Nt+1 – Nt) Rt K/2 K Nt

The Orange Roughy, Hoplostethus atlanticus, is a large deep-sea fish, formerly known as the Chilean Slimehead. It lives in cold deep water and requires 20+ years to reach sexual maturity. First harvested in New Zealand in 1978, its populations had already crashed by 1990. The US imports 8,620 tonnes (19 million pounds) per year. It is fished by trawling, which destroys 1-2 tonnes of coral reef for every tonne of fish caught. MSY ~ 7,500 tonnes.

The Grand Banks Fisheries Once had the world's most abundant populations of Atlantic cod. It collapsed in the 1990’s and a complete moratorium on fishing was imposed in 1994. Cod: A Biography of the Fish that Changed the World (1997 by Mark Kurlansky) North Atlantic Cod Gadus morhua

Atlantic Cod Marine Resources Service

The Grand Banks Fisheries Once abundant along East Coast from New Foundland to South Carolina, it is now threatened It is not hunted for food but its population has collapsed because it is part of the by-catch, i.e., other fish caught while trawling for food fish. Winter Skate Leucoraja ocellata

The Grand Banks Fisheries Striped Bass populations collapsed in the 1990’s but have since recovered. Spawning grounds in Chesapeake Bay are threatened as huge poultry farms feed Menhaden, their prey fish, to chickens and as poultry waste diminishes the habitat in the Bay. Striper Wars: An American Fish Story (2005 by D. Russell). . Striped Bass Morone saxatilis

Marine Resources Service

Tragedy of the Commons and Fisheries Management U.S. and World commercial fisheries are dominated by open access, the 1600’s law of freedom of the seas. The fundamental problem with managing the world's fisheries is the “tragedy of the commons”: Why should one individual or one country limit its fishing for the benefit of another individual or country? Since 1950, 29% of commercial fish species have suffered population collapse (a loss of 90% or more). If current trends are left in place, the last currently commercial fish species will be lost in 2048. Science 134:747-790 November 2006.

Human Population Growth: Plot of ‘r’ against Nt for Humans: Population Growth Rate per capita, r, has been increasing as human population size or density, Nt, increases: Positive Density Dependence. r Nt