IJPAM: Volume 60, No. 4 (2010)

EXTINCTION AND GROWTH IN EVOLUTIONARY
AND BIOLOGICAL SYSTEMS

William C. Troy
Department of Mathematics
University of Pittsburgh
Thack 503, Pittsburgh, PA 15260, USA
e-mail: [email protected]


Abstract.The consequences of global warming include predictions of 300,000 deaths per year, as well as the extinction of 100-500 species of birds per degree centigrade warming. Warming effects are also thought to play a role in runaway growth, e.g. the quagga mussel invasion of Lake Mead. At the same time the global financial community is suffering the effects of runaway growth (e.g. the housing market bubble), and the collapse of financial markets. The mechanisms underlying growth or extinction are poorly understood. We investigate these mechanisms in a scalar, continuous time equation which models the effects of environmental fluctuations. In addition to modeling biological populations, the model is the first component of Black Scholes stock price theory. In both settings the model predicts extinction of realizations when volatility exceeds a critical value. A major challenge is to estimate realizations at specific times during growth or extinction events. Thus, we derive dynamic bounds which estimate realizations during such events. These estimates give new insights into stock price collapse. We extend our results to a two-species system where the first is dominant, yet the second is less vulnerable to environmental fluctuations. When variation exceeds criticality the first species goes extinct, and the second emerges into a new state. We derive an estimate for the time lag between extinction and emergence. This estimate provides a step towards explaining the 2000 Kirchner-Weil discovery of a significant time lag between mass extinctions and repopulation.

Received: March 18, 2010

AMS Subject Classification: 34B15, 34C23, 34C11

Key Words and Phrases: Ito's Lemma, extinction, growth

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 4