Open access and extinction of the passenger pigeon in north america

Abstract

An open access model is formulated where X is a renewable resource and E is the level of effort devoted to harvest. Net growth is assumed to exhibit critical depensation and the open access system is described by two nonlinear differential equations. (formula presented) where r > 0 is the intrinsic growth rate, K1 is the minimum viable population level, K 2 is the environmental carrying capacity, K 2 > K 1 > 0, q > 0 is the catchability coefficient, ? > 0 is an adjustment coefficient, (p – s) > 0 is the market price net of shipping cost, and c > 0 is the unit cost of effort at the harvest site. It is shown that the E= 0 isocline is a vertical line at X∞=c/[(p- s)q] and that the open access system passes through a supercritical Hopf bifurcation as X∞ moves from a level above (K 1 +K 2 )/2 to a level below (K 1 +K 2 )/2. For X∞ above (K 1 +K 2 )/2 the open access equilibrium is locally stable. For X∞ below (K 1 +K 2 )/2 the open access equilibrium will be locally unstable. At X∞=(K 1 +K 2 )/2 the system has a stable limit cycle. This analysis is useful in interpreting the economic history of the passenger pigeon. The limited empirical evidence would suggest that X∞=c/[(p – s)q] declined below (K 1 +K 2 )/2 during the last half of the 19th century as a result of improved rail transport and communications (the telegraph). It is thought that the passenger pigeon was extinct in the wild by 1901. The last passenger pigeon died in captivity at the Cincinnati Zoological Gardens on September 1, 1914. © 2005 Rocky Mountain Mathematics Consortium.