Computerized simulation models have been developed to project abundances of insect populations affecting crop and forest resources (e.g., Gutierrez 1996, Royama 1992, Rykiel et al. 1984). The models developed for several important forest and range insects are arguably the most sophisticated population dynamics models developed to date because they incorporate long time frames, effects of a variety of interacting factors (including climate, soils, host plant variables, competition, and predation) on insect populations, and effects of population change on ecosystem structure and processes. Often, the population dynamics model is integrated with plant growth models; impact models that address effects of population change on ecological, social, and economic variables; and management models that address effects of manipulated resource availability and insect mortality on the insect population (Colbert and Campbell 1978, Leuschner 1980). As more information becomes available on population responses to various factors, or effects on ecosystem processes, the model can be updated, increasing its representation of population dynamics and the accuracy of predictions.

Effects of various factors can be modeled as deterministic (fixed values), stochastic (values based on probability functions), or chaotic (random values) variables (e.g., Croft and Gutierrez 1991, Cushing et al. 2003,Hassell et al. 1991, Logan and Allen 1992). If natality, mortality, and survival are highly correlated with temperature, these rates would be modeled as a deterministic function of temperature. However, effects of plant condition on these rates might be described best by probability functions and modeled stochastically (Fargo et al. 1982, Matis et al. 1994).

Advances in chaos theory are contributing to development of population models that more accurately represent the erratic behavior of many insect populations (Cavalieri and Kogak 1994, 1995a, b, Constantino et al. 1997, Cushing et al. 2003, Hassell et al. 1991, Logan and Allen 1992). Chaos theory addresses the unpredictable ways in which initial conditions of a system can affect subsequent system behavior. In other words, population trend at any instant is the result of the unique combination of population and environmental conditions at that instant. For example, changes in gene frequencies and behavior of individuals over time affect the way in which populations respond to environmental conditions. Time lags, nested cycles, and nonlinear interactions with other populations are characteristics of ecological structure that inherently destabilize mathematical models and introduce chaos (Cushing et al. 2003, Logan and Allen 1992).

Chaos has been difficult to demonstrate in population models, and its importance to population dynamics is a topic of debate. Dennis et al. (2001) demonstrated that a deterministic skeleton model of flour beetle, Tribolium castaneum, population dynamics accounted for >92% of the variability in life stage abundances but was strongly influenced by chaotic behavior at certain values for the coefficient of adult cannibalism of pupae (Fig. 6.8).

Several studies suggest that insect population dynamics can undergo recurring transition between stable and chaotic phases when certain variables have values that place the system near a transition point between order and chaos (Cavalieri and Kogak 1995a, b, Constantino et al. 1997) or when influenced by a generalist predator and specialist pathogen (Dwyer et al. 2004). Cavalieri and Kogak (1994, 1995b) found that small changes in weather-related parameters (increased mortality of pathogen-infected individuals or decreased natality of uninfected individuals) in a European corn borer, Ostrinia nubilalis, population dynamics model caused a regular population cycle to become erratic. When this chaotic state was reached, the population reached higher abundances than it did during stable cycles, suggesting that small changes in population parameters resulting from biological control agents could be counterproductive. Although chaotic behavior fundamentally limits long-term prediction of insect population dynamics, improved modeling of transitions between deterministic or stochastic phases and chaotic phases may facilitate prediction of short-term dynamics (Cav-alieri and Kogak 1994, Cushing et al. 2003, Logan and Allen 1992).

5-cycles 23.0%

Other cycles 37.9%

5-cycles 23.0%

Other cycles 37.9%

Other cycles 68.2%

Other cycles 68.2%

13-cycles 29.9%

Chaos 1.9%

13-cycles 29.9%

Chaos 1.9%

8-cycles 99.8%

8-cycles 99.8%

Chaos 83.5%

Chaos 83.5%

Other cycles 9.3%

19-cycles 7.1%

Other cycles 9.3%

19-cycles 7.1%

3-cycles 100.0%

3-cycles 100.0%

3-cycles 45.1%

3-cycles 45.1%

Frequency of predicted deterministic attractors for modeled survival probabilities of pupae in the presence of cannibalistic adults (cpa) of Tribolium castaneum for 2000 bootstrap parameter estimates. For example, for cpa = 0.35, 83.5% of estimates produced chaotic attractors, 7.1% produced stable 19-cycles, and 9.3% produced stable cycles of higher periods. From Dennis et al. (2001) with permission of the Ecological Society of America. Please see extended permission list pg 570.

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