9.4. Quantitative Measures of Stability

   In the [1]previous section we have discussed qualitative indicators of stability.
   According to these indicators, a model is either stable if its trajectory
   converges to an equilibrium state or unstable if it diverges from the equilibrium
   after small disturbances. However, real populations never converge to an
   equilibrium because of the random noise associated with weather and other
   stochastic factors. Thus, qualitative stability has a vague biological meaning.
   Ecologists are more interested in quantitative indicators of stability which
   represent the ability of the population to resist environmental fluctuations.

   Robert May (1973) suggested to measure system stability by the maximum real part
   of eigenvalues of the linearized model. It was shown that this value correlates
   with the variance of population fluctuations in stochastic models.

   Sharov (1991, 1992) suggested measures of m- and v-stability that characterize
   the stability of the mean (m) and variance (v) of population density (initially
   these measures were called as coefficients of buffering and homeostasis, see
   Sharov [1985, 1986]). Later they were re-invented by Ives (1995a, 1995b). They
   can be used to predict the effect of environmental changes (e.g., global warming
   or pest management) on the mean and variance of population numbers

   M-stability (MS) was defined as the ratio of the change in mean log population
   density, N, as a response to the change in mean value of some environmental
   factor, v.

                                      [eq16.gif]

   M-stability is the reciprocal of the sensitivity of mean population density to
   the mean value of factor v. Log-transformation of population density is important
   because it makes population models closer to linear.

   For example, if v is temperature which is going to change by 2 degrees due to
   global warming, and log population density per ha (log base e) will increase from
   1 to 1.5, then the sensitivity is S=(1.5-1)/2=0.25, and m-stability MS=4. The
   population with higher m-stability will change less than the population with low
   m-stability under the same changes in average factors.

   Strong population regulation increases m-stability because regulating mechanisms
   will resist to the changes in population density. Let's assume that regulation is
   caused by interspecific competition. Then, if conditions become favorable for the
   population, then the organisms will increase their reproduction rate. However, as
   population density increases, mortality due to competition increases too and
   partially compensates increased reproduction rates. If conditions become less
   favorable, then density will decline and mortality due to competition will
   decrease and partially compensate the decrease in reproduction rates.

   If population dynamics is described by a mathematical model then m-stability can
   be estimated from that model. The simplest example is the logistic model. Mean
   population density in the logistic model equals to carrying capacity, K. If the
   factor v affects K, then [eq17.gif] . If the factor v affects population growth
   rate, r, but does not affect carrying capacity, then mean population density will
   not respond to factor change, and thus, m-stability will be infinitely large.

   V-stability (VS) was defined as a ratio of the variance of additive random noise
   [eq12ksi1.gif] to the variance of log population numbers [eq12ksi2.gif] :

                                      [eq12.gif]

   Population that has smaller fluctuations of population numbers than another
   population that experience the same intensity of additive environmental noise has
   a higher v-stability. To estimate v-stability in the [2]Ricker's model we can use
   the linearized model at the equilibrium point:

                                      [eq13.gif]

   where N is log population density, and [eq13ksi2.gif] is the white noise with a
   zero mean. Noise is not correlated with log population numbers. Thus:

                                      [eq14.gif]

                                     [gvstab.gif]

   This graph shows that v-stability equals to zero at r=0 and r=2 (these are the
   boundaries of quantitative stability). V-stability has a maximum at r = 1.

   References

   Ives, A.R. 1995. Predicting the response of populations to environmental change.
   Ecology 76: 926-941.
   Ives, A.R. 1995. Measuring resilience in stochastic systems. Ecol. Monogr. 65:
   217-233.
   May, R.M. 1973. Stability in randomly fluctuating versus deterministic
   environments. Amer. Natur., 107: 621 650.
   Sharov, A.A. 1985. Insect pest population management taking into account natural
   mechanisms of population dynamics. Zoologicheskii Zhurnal (Zoological Journal),
   64: 1298 1308 (in Russian).
   Sharov, A.A. 1986. Population bufferity and homeastasis and their role in
   population dynamics. Zhurnal Obshchej Biologii (Journal of General Biology), 47:
   183 192 (in Russian).
   Sharov, A.A. 1991. Integrating host, natural enemy, and other processes in
   population models of the pine sawfly. In: Y.N.Baranchikov et al. [eds.] Forest
   Insect Guilds: Patterns of Interaction with Host Trees. U.S. Dep. Agric. For.
   Serv. Gen. Tech. Rep. NE-153. pp. 187-198.
   Sharov, A.A. 1992. Life-system approach: a system paradigm in population ecology.
   Oikos 63: 485-494. [3]Get a reprint! (PDF)

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   [7]Alexei Sharov 1/12/96

References

   1. http://alexei.nfshost.com/PopEcol/lec9/equilib.html
   2. http://alexei.nfshost.com/PopEcol/lec5/explog.html#ricker
   3. http://alexei.nfshost.com/~sharov/pdf/lifesys.pdf
   4. http://alexei.nfshost.com/PopEcol/lec9/equilib.html
   5. http://alexei.nfshost.com/PopEcol/lec9/stabil.html
   6. http://alexei.nfshost.com/PopEcol/lec9/chaos.html
   7. http://alexei.nfshost.com/~sharov/alexei.html