Insects outbreaks

For a biological example of bifurcation and catastrophe, we turn now to a model for the sudden outbreak of an insect called the spruce budworm. This insect is a se- 

Ludwig et al. (1978) proposed and analyzed an elegant model of the interaction between budworms and the forest. They simplified the problem by exploiting a separation of time scales the budworm population evolves on a fast time scale (they can increase their density fivefold in a year, so they have a characteristic time scale of months), whereas the trees grow and die on a slow time scale (they can completely replace their foliage in about 7-10 years, and their life span in the absence of budworms is 100-150 years.) Thus, as far as the budworm dynamics are concerned, the forest variables may be treated as constants. At the end of the analysis, we will allow the forest variables to drift very slowly—this drift ultimately triggers an outbreak. 

We now have several questions to answer. What do we mean by an outbreak in the context of this model The idea must be that, as parameters drift, the bud- 

The model (1) has four parameters R, K, A, and B. As usual, there are various ways to nondimensionalize the system. For example, both A and K have the same dimension as A, and so either N/A or N/K could serve as a dimensionless population level. It often takes some trial and error to find the best choice. In this case, our heuristic will be to scale the equation so that all the dimensionless groups are pushed into the logistic part of the dynamics, with none in the predation part. This turns out to ease the graphical analysis of the fixed points. 

Equation (2) suggests that we should introduce a dimensionless time t and dimensionless groups r and k, as follows  

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The documentation of regional level terrestrial consequences of acid deposition is complicated. For example, forested ecosystems m eastern North America can he influenced by other factors such as high atmospheric ozone concentrations, drought, insect outbreaks and disease, sometimes from non-native sources. However there is a general consensus on some impacts of acidic depositon on both soils and forests m sensitive regions. 

Scriber, J.M. Halnze, J.H. In "Insect Outbreaks Ecological and Evolutionary Processes" Barbosa, P. Schultz, J.C., Eds. Academic Press New York, 1986, In press. 

An Insect outbreak, therefore, would not be controlled on a site being managed for wildlife If the reduction In the timber resource Improved the site for wildlife without greatly damaging other resource values. 

Table II shows pesticides used by the US Forest Service and the amounts used in 1980. On this list 2,4-D is in first place, being used at in an amount of 215,000 pounds per year. In 1980 the second most commonly used pesticide by the USFS was the insecticide malathion at in an amount of 102,000 pounds. There are only three insecticides on this list of the nine most commonly used pesticides in the USFS. The insecticide use rate will vary considerably from year to year as its use is dependent on insect outbreaks, whereas herbicides are used at a more constant rate because the appearance of weeds and brush, as they affect forest management, do not occur as periodic outbreaks.

Pesticide use in private forestry will more-or-less parallel that in the USFS, with one exception. They will not use malathion, carbaryl, or azinophos-methyl, the insecticides shown in Table II, because nearly all insect outbreaks are managed by either a federal or other public agency rather than by private forestry. Although 2,4-D is the principal pesticide used in forest management, Table III shows that its use in Oregon for 1979 in forestry is only about 10 percent of the total use in the State. The principal use is in wheat and other grains, which utilized nearly 3/4 of a million pounds, while forestry used 147,000 pounds. 

The term catastrophe is motivated by the fact that as parameters change, the state of the system can be carried over the edge of the upper surface, after which it drops discontinuously to the lower surface (Figure 3.6.6). This jump could be truly catastrophic for the equilibrium of a bridge or a building. We will see scientific examples of catastrophes in the context of insect outbreaks (Section 3.7) and in the following example from mechanics. 

Warm-up question about insect outbreak model) point X = 0 is always unstable for Equation (3.7.3). 

Ludwig, D., Jones, D. D., and Holling, C. S. (1978) Qualitative analysis of insect outbreak systems the spruce budworm and forest. J. Anim. Ecol. 47, 315. 

A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory. 

Ewald PW. Pathogen-induced cycling of outbreak insect populations. In Barbosa P, Schultz JC, eds. Insect Outbreaks. New York Academic Press, pp 269-286, 1987. 

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