First and Second Order Dynamics

Feedback loops that may be characteristic of density-dependent mechanisms can be broken down into having first- or second-order dynamics. First-order dynamics occur over on a short timescale, with short lags (1-2 years) between the response of the herbivore population and the feedback mechanism. Second-order dynamics have pronounced time lags (3-5 years) between the mechanism and the effects on the herbivore population. Second-order dynamics can cause cycles, but may also result in more complex phenomena, such as chaotic (very dependent on initial conditions) and bounded fluctuations in population density. In general, ecologists believe that population cycles are caused by second-order or higher dynamics, since an appreciable time lag between the factor involved and the herbivore density is needed to produce periodicity in herbivore populations.

Some properties of herbivore populations themselves may cause negative feedback loops and thus may contribute to the maintenance of population cycles by causing or exacerbating the decline portions of the cycles. For example, increasing population densities may cause more competition for food or more aggression among conspeci-fics, leading to increased rates of mortality or decreased birth rates. Increased competition for shelter or mating sites may result in many individuals succumbing to prey or failing to recruit offspring into the next generation. These mechanisms, however, may not be sufficient to cause cycling, since there may be only short or even nonexistent time lags between consumption of food or space and subsequent reductions in herbivore densities. One prominent exception is the population cycles of Soay sheep populations on the St. Kilda archipelago off the coast of Scotland. This population goes through abrupt and somewhat regular fluctuations in number, and has been well documented for over 30 years. There are no predators or significant parasites on these islands, so starvation and regrowth of forage are thought to be the main driver ofthe cycles. Some recent analyses also conclude that growth of forage and concomitant starvation is also affected by climatic variables such as the North Atlantic Oscillation (NOA).

For herbivores, like the Soay sheep, an obvious limiting factor in the growth and maintenance ofpopulations is their food sources - plants. Population cycles of herbivores could be maintained if the plants that they feed on also undergo large fluctuations in their population densities. For example, defoliation of trees often reduces the quality and increases toughness of vegetation in following seasons. This phenomenon, coupled with the intrinsic rate of population growth of the herbivores, may result in a feedback loop in the population size of the herbivore. This feedback can be immediate or delayed, and thus characteristic of first- or second-order dynamics, depending on the particular system involved.

If plants were the sole determining factor in the cycling process, then herbivore densities should mirror closely the densities of their host plants with some time lag, and would thus have low-order dynamics (zero or first-order) This explanation, however, has not been supported by data since plant densities rarely undergo such drastic cycles as herbivores in any system. For example, the larch budmoth (Zeiraphera diniana) (LBM), which cycles approximately every 9 years, plant mortality is only around 1% after attack, and plant biomass is only reduced by 50% during the outbreaks. Thus, it seems unlikely that the drastic crashes in herbivore densities are due to a lack of food resource in this system.

Similarly, in a large data set of Fennoscandian voles, oscillations in population density are probably not caused by concomitant oscillations in plant abundance alone. Oscillations in these populations have characteristics of second-order dynamics with relatively long lags in response time. Again, if determining factor was available plant biomass for consumption, then there will be little time lag between the action of the herbivores and the decline in herbivore population densities.

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