Most species live in species-rich food webs; yet, for a century, most mathematical models for population dynamics have included only one or two species1–3. We ask whether such models are relevant to the real world. Two-species population models of an · Single-species models for many-species food webs. Murdoch WW(1), Kendall BE, Nisbet RM, Briggs CJ, McCauley E, Bolser R. Author information: (1)Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara, California , USA. murdoch@blogger.com Most species live in species-rich food webs; yet, for a century, most mathematical models Cited by: one-species model is a valid representation for generalist popu- lation dynamics in many-species food webs. Conventional approaches to nonlinear time-series analysis focus
Single-species models for many-species food webs | Nature
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A short summary of this paper. Single-species models for many-species food webs. letters to nature species single species models for many species food webs a system9, These are powerful methods, but require time series much longer than those typically available for field popu- Single-species models for lations Theory for stage-structured populations leads us to consider three classes of W.
Single-species populations with direct density dependence in E. approximately constant resource recruitment in these consumer— Most species live in species-rich food webs; yet, for a century, resource models induces weak coupling and a collapse to single- most mathematical models for population dynamics have species dynamics; direct or delayed density dependence in the included only one or two species1—3. We ask whether such models consumer can then produce single-generation cycles or delayed- are relevant to the real world.
Two-species population models of feedback cycles in the consumer, with the period determined by an interacting consumer and resource collapse to one-species consumer development time4—6. dynamics when recruitment to the resource population is unre- We first establish that specialist consumer—resource cycles can be lated to resource abundance, thereby weakening the coupling distinguished from single-species cycles by their scaled periods between consumer and resource4—6.
We predict that, in nature, cycle period divided by time to maturity. If TC and TR are the generalist consumers that feed on many species should similarly maturation times of the consumer and resource respectively, then show one-species dynamics.
We test this prediction using cyclic cycles in single-species models have periods that seldom exceed 4TC, populations, in which it is easier to infer underlying mechan- as noted, whereas consumer resource cycles have periods seldom isms7, and which are widespread in nature8. Here we show that less than 4TC þ 2TR Box 1. one-species cycles can be distinguished from consumer—resource Next, the collapse from consumer-resource dynamics to single- cycles by their periods, single species models for many species food webs.
We then analyse a large number of time species dynamics caused by weak coupling suggests the following series from cyclic populations in nature and show that almost all prediction. Generalist consumers should typically be weakly cycling, generalist consumers examined have periods that are coupled to any one of their prey populations because, when feeding consistent single species models for many species food webs one-species dynamics.
Thus generalist consu- on many different species, they cannot be strongly coupled to any mers indeed behave as if they were one-species populations, and a one of them.
In particular, total resource recruitment rate will be one-species model is a valid representation for generalist popu- largely independent of the abundance of the consumer and of any lation dynamics in many-species food webs. particular resource population. We therefore predict that, among Conventional approaches to nonlinear time-series analysis focus cyclic species, generalists should typically show single-species cycles.
on dynamical invariants such as the dimension of the series, which Specialist species, on the other hand, should more typically show has recently been used to infer the number of strongly interacting consumer—resource cycles, although strong density dependence Table 1 Species analysed Generalists Specialists Asterisk indicates zero series in the class. a, Number Pterostichus versicolor, carabid beetle Vespula sp.
b, Distribution of cycles among classes Except as noted, generalists had single-species cycles and specialists had consumer—resource defined by scaled period.
No theoretical models Methods and Supplementary Information 3. All appropriate series for a species were analysed Methods and, if this exceeded 1, the number of series analysed is in parenthesis. cycles fall in the intermediate class INT between single-species and consumer—resource monacha and two of the mink series showed single-species cycles.
Each of the two carabid cycles. beetle species showed consumer—resource cycles. NATURE VOL 30 MAY www. A lower bound for periods of consumer—resource cycles We analysed time series from natural populations compris- ing 40 specieseach of which exhibited statistically significant A family of consumer—resource models has the property that periodic or cycles and met criteria ensuring a rigorous test of the predictions quasi-periodic solutions have period greater than six These models Methods.
We classified each species into one of two trophic roles assume that both the consumer and resource species reproduce only Methods. Here we generalize the feed on many resource species over their lifetime and in turn are not work of ref. We then estimated cycle periods and strict mathematical conditions for that bound are relaxed.
calculated scaled periods, single species models for many species food webs. Our focus was on generalists, because the We consider a tightly coupled consumer—resource model of the specialists in the database are known, in general, to have long-period general form: cycles. We used the distribution of specialist cycle periods as a basis Rt ¼ lRt21 fðRt21 ; Ct21 Þ þ SR Rt21 ð1Þ for comparison and to test the Box 1 prediction that scaled long Ct ¼ lRt21 ð1 2 fðRt21 ; Ct21 ÞÞ þ SC Ct21 ð2Þ periods are not less than 4TC þ 2TR.
single species models for many species food webs R t and C t represent the density of adults in the resource and Our predictions are well supported. The generalists and special- consumer populations in year t, l is single species models for many species food webs geometric rate of resource ists did not separate cleanly when we examined period measured in increase, f R t,C t is the fraction of the resource population that years mainly because a substantial fraction of generalist periods are survives attack by the consumer, and S R, S C are the fractions of the longer than four, or even six, years; see Fig.
almost perfectly when we used the scaled period Fig. The characteristic equation single species models for many species food webs the conditions for local including those with periods greater than four years.
There are no stability of the equilibrium is: cycles with intermediate scaled period, that is, between 4TC and m 2 þ A1 m þ A2 ¼ 0 ð3Þ 4TC þ 2TR, in spite of the fact that many periods lie between four and six years Fig. As expected, the great majority of cyclic series with A1 ¼ 21 2 a 2 b 2 SC ; A2 ¼ aðl þ SR Þ þ SC ð1 þ bÞ; where a ¼ from specialists are true consumer—resource cycles Fig.
respect to i for i ¼ R or C and m is an eigenvalue. With forms of the Furthermore, as predicted by the theory discussed above, when functional response most appropriate for specialist consumers types specialists have shorter periods only three casessingle species models for many species food webs, they are single- 1 or 2a and b are both non-negative.
species periods rather than intermediate periods Fig. On the local stability boundary, A 2 ¼ 1, and the period of the cycles Single-species cycles occur in a broad taxonomic range but are is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
especially prevalent in fish Table 1. With some additional algebra, it can be groups that were excluded because the species have multiple shown that the cycle period approaches six for large l only if b ¼ 0 generations per year.
If either b. Cycle period increases as l decreases from species in many-species food webs, but that prey species diversity, the limit to lower, and biologically realistic, values. a prevalent feature of natural systems, itself facilitates such simpli- Resource density dependence can reduce the period, but only fication.
Remaining challenges include adding taxonomic variety to slightly. We analysed a variant of the above model with the standard the database, developing appropriate theory on cycle periods for Nicholson—Bailey assumption, f ¼ exp 2aC timplying b ¼ 0, but with multivoltine species, and determining whether the simplification either Ricker or Beverton—Holt density dependence in the resource applies in non-cyclic generalist populations.
A population. A period of six is no longer a lower bound, but we found that consumer—resource periods always exceeded 5. Delayed-feedback cycles a period of two arise in the information Supplementary Information 2. Criteria for excluding time series of field Ricker version with sufficiently strong resource density dependence.
respectively, has equations: 2 There is strong evidence the population has no causal role in the cycle 43 series from Rt ¼ lRt2T R fðRt2T R ; Ct2T R Þ þ SR Rt21 ð5Þ high-latitude species, mainly foxes, entrained by the lynx—snowshoe hare cycle or, less often, Ct ¼ lRt2T C ð1 2 fðRt2T C ; Ct2T C ÞÞ þ SC Ct21 ð6Þ by microtine voles, lemmings, and so on cycles; these populations track but do not cause the cycles19— Mink and muskrat series were analysed because there is evidence that they are With the same restrictions as before b ¼ 0 and S C ¼ 0and one causally involved in a distinct cycle19, as were the mountain and brown hare series An outline of the less than one year and breeding is more frequent than annual.
Theory on cycle periods is not available for seasonally driven, properly age-structured, consumer—resource models proof is in Supplementary Information 1. We have not managed to of species with more than one generation per year. This excludes 31 series, mainly from prove that this limit is a lower bound as was true with a period of six, voles.
but extensive numerical studies suggest that it is. any results in which the estimated period was more than one-third of the series length. To avoid biologically meaningless results, we only analysed data with mean counts of more However, such calculations are typically a good indicator of the periods than 10 individuals and cycles in which the maximum population was on average at least of all but very large-amplitude quasi-cycles or limit cycles.
With large- twice the minimum. Generalist consumers—usually top predators with broad diets—could typically be categorized easily and unambiguously. Lepidopteran forest pests were as a lower bound.
classified as specialist prey of parasitoids 23, although some cycles may be specialized © Nature Publishing Group NATURE VOL 30 MAY www. The last would increase the The authors declare that they have no competing financial interests prevalence of single-species dynamics. Scaled period, t, was calculated by dividing the period in years by time to maturity in Correspondence and requests for materials should be addressed to W.
yearsTC sources in Supplementary Information 3. For cases in which t. asked if the cycle period in years exceeded 4TC þ 2TR ; in series involving lynx average maturation time ¼ 1. Because maturation times and cycle periods are approximations, estimates of t were rounded to the nearest integer. Periods with t ¼ 1 were assigned to single-generation cycles; those with 2 t 4 were assigned to delayed- feedback cycles.
We used multitaper spectral analysis25,26 using the algorithm described in ref, single species models for many species food webs.
This technique avoids the often arbitrary choice of data tapering by using a series of optimal Precise inhibition single species models for many species food webs essential for slepian tapers.
Food Chains \u0026 Food Webs - Ecology \u0026 Environment - Biology - FuseSchool
, time: 2:55(PDF) Single-species models for many-species food webs | Cheryl Briggs - blogger.com
Most species live in species-rich food webs; yet, for a century, most mathematical models for population dynamics have included only one or two species. We ask whether such models are relevant to the real world. Two-species population models of an interacting consumer and resource collapse to one-species dynamics when recruitment to the resource population is unrelated to resource · Single-species models for many-species food webs. Murdoch WW(1), Kendall BE, Nisbet RM, Briggs CJ, McCauley E, Bolser R. Author information: (1)Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara, California , USA. murdoch@blogger.com Most species live in species-rich food webs; yet, for a century, most mathematical models Cited by: one-species model is a valid representation for generalist popu- lation dynamics in many-species food webs. Conventional approaches to nonlinear time-series analysis focus
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