IJPAM: Volume 60, No. 4 (2010)
AND BIOLOGICAL SYSTEMS
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