OfRo

The model of Leslie gives an accurate estimation of r. Method #1

For simplicity we take years as time units. However, the same logic can be applied to days or weeks.

The number of organisms in age x in year t is equal to the number of new-born organisms (x=0) x years ago multiplied by their survival (lx) till age x:

Now, equation [2] of the Leslie model can be re-written as:

After initial damped fluctuations, the Leslie model shows exponential growth and age distribution stabilizes. Thus, the number of organisms in any age class will grow exponentially. In particular, the number of new-born organisms increases exponentially:

No^ No^airtr-cx+lfl

Then,

Now we combine equations [4] and [5]:

The term N0,t-x can be taken out of the summation expression and then we get the equation:

Equation [6] can be used to estimate r. The sum at the left side can be estimated for different values of r, and then we can select the r-value that makes this sum equal to 1. It makes sense to start with the r-value estimated using the approximate method discussed in the previous chapter. Thus, we start with r=0.181 and get the sum [6] equal to 0.9033. When r increases, then the value of the sum [6] decreases because r is included as a negative exponent. Obtained value of the sum appeared to be less than 1, and thus, we need to try smaller values of r. Let's select r=0.16. Then the sum is equal to 1.0092. The exact value of r can be found by linear interpolation:

0.185 Oil F

Now we check the solution: when r=0.1617 then the sum [6] is equal to 1.00014 which is very close to 1.

Note: In Pielou (1978), this example is estimated in a different way which is difficult to understand. She constructed a different matrix by adjusting reproduction rates and provided no explanation for this adjustment.

Method #2

Intrinsic rate of population increase can be estimated as the logarithm of the only real and positive eigenvalue of the transition matrix. The theory of eigenvalues is the central topic in linear algebra. It is used to reduce multidimensional problems to one-dimensional problems. I recommend to study this topic for those students who plan to be quantitative ecologists. Here we will only estimate the eigenvalue using available software without going into details of the algorithm. The only real and positive eigenvalue of our matrix is equal to X=1.176. Then, r = ln(k) = 0.162 which is very close to the value estimated by method #1.

Checking the result

Obtained value of r can be checked by estimating the regression of log population numbers versus time. Initial years should be ignored because age structure has not been stabilized yet. The slope of this regression should be equal to r. If we take the time interval from t = 25 to 50, then the regression equation is ln(N) = 4.3557 + 0.1617 t. Regression slope is exactly equal to r estimated by method #1.

7.4. Estimation of stable age distribution

Equation [5] can be re-written as:

Substituting this equation into [3] we get the relationship between the number of organisms in age x and in age 0 in a stable age distribution:

Now we can estimate the proportion of organisms, c , in age x:

Age, x

lx

exp(-rx)

lxexp(-rx)

cx

Simulated cx

0

1.000

1.0000

1.0000

0.2413

0.2413

1

0.845

0.8507

0.7188

0.1734

0.1734

2

0.824

0.7237

0.5963

0.1439

0.1439

3

0.795

0.6156

0.4894

0.1181

0.1181

4

0.755

0.5237

0.3954

0.0954

0.0954

5

0.699

0.4455

0.3114

0.0751

0.0751

6

0.626

0.3790

0.2373

0.0572

0.0572

7

0.532

0.3224

0.1715

0.0414

0.0414

8

0.418

0.2743

0.1147

0.0277

0.0277

9

0.289

0.2333

0.0674

0.0163

0.0163

10

0.162

0.1985

0.0322

0.0078

0.0078

11

0.060

0.1689

0.0101

0.0024

0.0024

Total

4.1445

1.0000

1.0000

Age distribution estimated using equation [7] (column 5) coincided with simulated age distribution after 50 iterations of the model.

7.5. Modifications of the Leslie model

1. Variable matrix elements. Survival and reproduction rate of organisms may depend on a variety of factors: temperature, habitat characteristics, natural enemies, food, etc. To represent these dependencies, the elements of Leslie model can be replaced by equations that specify survival and reproduction rates as functions of various factors. Equations can be obtained from experimental data.

2. Distributed delays. Age and time are equivalent in the original Leslie model, and thus, all organisms develop synchronously with constant rate. However, development rate of invertebrates and plants is not constant: it depends on temperature and may vary among individuals. Individual variation of development rates is called "distributed delay" because there is a distribution of time when organisms reach maturity. Transition matrix can be modified to incorporate these features.

This matrix has non-zero diagonal elements, and thus, some proportion of organisms remain in the same age class when time increases. As a result, development goes slower than it would be in the original Leslie model. The rate of development can be adjusted by changing relative values of diagonal and sub-diagonal elements

This matrix allows organisms to leap over several age intervals in one time step. As a result, development goes faster than it would be in the original Leslie model. The rate of development can be adjusted by changing the length of age leaps.

3. Partitioning the life cycle into stages. Many invertebrate species have a complex life cycle that includes several stages. For example, holometabolous insects usually have 4 stages: egg, larvae, pupae, and adult. Each of these stages may include several age intervals: In these models, age is no longer measured in calendar time units (e.g., days or years). Instead, it is measured in independent units which can be interpreted as "physiological age". The concept of physiological age will be discussed in details in the next chapter. It can be used to define "rate of development" as the average increment of physiological age per calendar time unit.

Lecture 8. Development of poikilothermous organisms, degree-days

8.1. Rate of development

Homeothermous organisms are warm blooded (mammals, birds) Poikilothermous organisms are cold blooded (all invertebrates, plants, fishes, amphibians, reptiles)

Development of poikilothermous organisms depends on temperature, whereas development of homeothermous organisms is temperature-independent.

The rate of development can be measured by a reciprocal of the number of time units (e.g., days) that is required for completion of development. Rates of development can be estimated for the entire onthogenesis or for a specific stage. For example, if it takes 15 days for an insect to develop from egg hatch till pupation, then the rate of larval development is v = 1/15 = 0.0667.

Temperature (°C), t

Development time,

T

Development rate, v = 1 / T

5

-

-

10

200

0.005

15

100

0.010

20

60

0.017

25

40

0.025

30

30

0.033

35

35

0.028

Here is the graph of development rate versus temperature:

Here is the graph of development rate versus temperature:

Taqnnlin, I fC)

In the temperature range from 10 to 30 (°C), development rate changes almost linearly with increasing temperature. At very low temperature there is no development, and at very high temperature development is retarded.

8.2. Simple degree-day model

Degree-day model is based on the assumption that development rate is a linear function of temperature.

In most cases, real development rate is indeed a linear function in the region of moderate temperatures (15-25°) (see figure above). Deviating points at temperatures that are too low or too high can be ignored. For example, in the figure, regression line is plotted for points from 10 to 30°. At 5° all organisms died and there was no development. At 35° organisms were overheated and development rate of survivors was reduced. If we are interested in simulating organism development in moderate temperatures, then the degree-day model will be the best choice.

Terms:

• tmin is the lower temperature limit; this is the temperature at which development rate reaches zero.

• S are degree-days; this is the effective temperature multiplied by the number of days required to complete development.

Lower temperature limit and degree-days can be estimated from the regression line of development rate versus temperature. We assume that regression equation is:

a + btmin = 0, by the definition of the lower temperature limit. Thus,

Duration of development is T = 1/v. Thus, degree-days can be estimated as:

This equation shows that degree-days do not depend on temperature! This is the principal feature of the degree-day model. In our case, S = 1 / 0.00142 = 704 degree x day. The units of degree-days are degrees (centigrade) multiplied by days. Some times, it is better to use degree-hours if development is very fast.

Because degree-days are temperature-independent, it is possible to use them to predict development time in experiments with variable temperature. In this case effective temperatures are accumulated day by day, and when the sum reaches S, then development is finished.

Example: tmin = 10 , S = 100.

Day No.

Average Temperature t

Effective temperature, t - tmin

Accumulated degree-days

1

15

5

5

2

18

8

13

3

25

15

28

4

23

13

41

5

24

14

55

6

18

8

63

7

17

7

70

8

15

5

75

9

18

8

83

10

15

5

88

11

22

12

100

12

25

15

115

Accumulated degree-days reach the value S = 100 on day 11. Thus, it takes 11 days to complete the development.

8.3. How to measure temperature?

Yes, use a thermometer! The only problem is when and where to measure it. The simplest way is to measure minimum and maximum temperatures each day and then take the average. Majority of weather data bases contain minimum and maximum temperatures. More accurate estimates can be obtained if temperature was measured several times a day at regular intervals (e.g. every 3 h). Then, you can average these measures.

The next problem is where to measure. It is obvious, that temperature should be measured where studied organisms are located. For example, if we study development of soil insects [e.g., wireworms (Elateridae)], then temperature should be measured in the soil at ca. 5-10 cm from the surface. Spider mites live on the lower leaf surface where temperature is several degrees lower than the ambient temperature. In this case, temperature should be measured under the leaf. Some times it is possible to build a regression model that predicts the temperature in a specific niche from ambient temperature recorded at weather stations.

Can we average temperature prior to analysis?

If development rate is a linear function of temperature, then temperature can be averaged prior to analysis. However, if this function is non-linear (e.g., in the improved degree-day model, see below), then temperature averaging may result in substantial errors.

8.4. Improved degree-day model

The purpose of modification is to use the model for a wide range of temperatures. Effective temperature (ET) is defined as follows:

This model is non-linear because the graph is not straight (see the figure above). Thus, temperature cannot be averaged! In particular, you cannot use average daily temperature. Instead, it is necessary to use actual temperature dynamics. Accumulated degree-days are equal to the area under the temperature curve restricted to the temperature interval between tmin and tm fmax-

fmax-

Light-blue area equals to accumulated degree-days. Here the average temperature is below tmin but organisms can accumulate some degree-days because daily maximums are above tmin.

Another example:

tine, days

Light-blue area again equals to accumulated degree-days. Daily maximum temperature exceeds tmax however this excess does not count in the accumulation of degree-days.

Non-linear models of development rate require simulation of diurnal temperature change. In most cases, only daily minimum and maximum temperature are known. There are two most frequently used methods for simulation temperature change. "Rectangular" model assumes that temperature stays at maximum for half of the day and stays at minimum for the other half:

This method is not very accurate but it is simple and fast.

"Sine-wave" method was developed by Allen (1976; Environ. Entomol. 5: 338-396). This model generates smooth temperature changes as in the second figure above. In the paper of Allen you will find a FORTRAN code that can be used for estimation of degree-days.

8.5. Other non-linear models of development

Other non-linear models of development

Improved degree-day model inherited a linear relationship between development rate and temperature (between two temperature limits tmin and tmax) from the simple degree-day model. Now it is time to make the next logical step: not to use linear relationships at all. Below two alternative models are shown: the logistic model and the normal distribution:

Non-linear models like the logistic model or the normal distribution are easy to interpret for experiments with constant temperature. In this case, development rate can be defined as a reciprocal of development time (v = 1/T). But what is development rate if temperature changes? We cannot accumulate degree-days any more because the model is non-linear. In attempt to answer this question a new concept has been developed: a concept of physiological time.

8.6. Physiological time

What is time? This is the most fundamental question in science. It attracted the best brains in human history from St. Augustin to Einstein. What is the relationship between time and dynamics? Do we measure dynamics relative to time, or may be time should be measured with dynamics? Newtonian physics considered absolute and universal time which was above nature. Alternative point of view is that each system has its own individual time measured by the number of events. For example, time in human life can be measured by the number of decisions made. A person who lives passively, and avoids making decisions, remains a child psychologically even if he is biologically an adult. Further discussion on the problem of biological time can be found in the paper of Sharov (1995).

We can apply the concept of biological time to the development of poikilothermous organisms. The progress in organism's development can be viewed as a biological clock that measures physiological time. Physiological time goes fast when it is warm and slow when it is cold. Then, development rate can be defined as advance of physiological time per unit of calendar time. For example, if development rate of insect larvae is 0.07 per day then, taking the entire physiological time at larval stage as 100%, in one day the insect will increment its physiological time by 7%.

To determine the duration of development we will accumulate development rates rather than effective temperatures. Development completes on the day when accumulated development rates reach 1 (=100%), i.e., the following equation is true:

where v(t) is the rate of development as a function of temperature, t; t(x) is temperature as a function of time, x; and T is the development time. If temperature is defined as a function of time (see the graph below), then at each time the rate of development is estimated as a non-linear function of time. Finally the integral of function v(t(x)) (the area under the curve) from 0 to T should be equal to 1.

Tme.xftiys)

The discrete-time version of the previous equation is:

There is no mechanical watch that measures physiological time. Also there are no hour and minute hands inside an organism. However, in some cases it is possible to find indicators of physiological time, e.g., concentration of hormones and metabolites, CO2 emission rate, etc. However, these indicators are usually not very accurate. For example, the respiration rate is high in gypsy moth eggs for ca. 2 weeks after oviposition; then it decreases and stay at low level. Thus, it is possible to distinguish between these two periods of egg development.

The problem of measurement can be evaded by preparing organisms of specific physiological age for the experiment. This is analogues to quantum mechanics where it is impossible to measure exact location of an electron without changing its momentum. But it is possible to prepare electrons with specific characteristics. To prepare organisms in a specific physiological age, we need to keep them in standard conditions for a specific time. For example, if the duration of insect pupal stage is 10 days in 25°, then after 5 days in this temperature, pupae will be in the middle of their pupal stage.

What are the advantages of physiological time concept?

As compared to the degree-day model, the concept of physiological time has the following advantages:

• It makes no arbitrarily assumptions on the shape of development rate function. This function can be linear, logistic, normal distribution, etc.

• It admits the effect of other factors besides temperature on the rate of development. The rate of development may depend on food quality and quantity, photoperiod, and other factors.

Before the concept of physiological time was developed there were attempts to incorporate the effect of factors other than temperature into the degree-day model. It was assumed that additional factors change the amount of degree-days required to complete development. For example, gypsy moth larvae develop faster on preferred tree species (oaks) as compared to non-preferred species (e.g., maples). It was assumed that more degree days should be accumulated by gypsy moth larvae on maple than on oak trees to pupate. This model does not work if larval migration is considered among trees because degree-days accumulated on different tree species are not comparable. For example, if larvae move from an oak tree to a maple tree, they become "younger". If they were ready to pupate on oak before migration, then after migration to maples they still have to accumulate additional degree-days to pupate.

8.7. How to combine physiological time with the model of Leslie?

The model of Leslie can be easily modified to incorporate physiological time. In the original model age and time were equivalent because each time step organisms were moving to the next age class. Now age will be measured in units of physiological time! Each column and row in the matrix corresponds to a specific physiological age.

Progress in physiological age in one time step may depend on temperature (3 arrows at the left side of this graph).

Also, individual variation in development rate (distributed delays) can be taken into account (branching arrow at the right side of the graph).

At the start of simulation all organisms can be placed into the first age class. Another option is to add variability in the starting date of development. For example, if we simulate insect larval development, it is unrealistic to assume synchronous egg hatch in one day. It is better to assume distributed egg hatch time. Three kinds of distributions are used most often: normal, logistic, and Weibul.

Cumulative distributions are described by equations:

Two graphs below show how to convert cumulative probability function of egg hatch time into the proportion of eggs that hatch each day (logistic and Weibull distributions):

Normal and logistic distributions are both symmetrical and are very similar. But Weibull distribution is asymmetrical. Actual distribution of egg hatch time is often asymmetrical, and thus, the Weibul distribution is usually better than the normal and logistic distributions.

8.8. Questions and Assignments to Lecture 8.

8.1 Insect development rate. Development of pea weevil (Bruchuspisorum) eggs was studied in the laboratory at constant temperatures (Smith, A.M. 1992. Environ. Entomol. 21:314-321):

Temperature, C

Egg development time, days

10.7

38.0

14.4

19.5

16.2

15.6

18.1

9.6

21.4

9.5

23.7

7.3

24.7

4.5

26.9

1. Plot development rate vs. temperature (use Excel)

2. Use linear regression to estimate lower temperature threshold and degree days required for egg development.

3. Average temperature in a sequence of days was: 15 20 25 20 15 10 15 10 15 20 15 20 15 10 15 20 25 20 15 20

4. When do you expect eggs to hatch if they were laid on the first day? (Note: ignore diurnal temperature change).

Lecture 9. Stability, Oscillations and Chaos in Population Dynamics

9.1. Introduction. Main Problems

Some populations have considerable oscillations in their numbers. There are well-known cycles in rodent populations. Many insect pests have regular or irregular outbreaks, e.g., larch budmoth in Switzerland, spruce budworm in Canada, gypsy moth in Europe and in USA.

Example: Gypsy moth population dynamics in Yugoslavia:

Gypsy moth numbers increased by several orders of magnitude. Pest outbreaks resulted in forest defoliation in large areas.

Main Problems:

• To evaluate population stability using models

• To examine effects of different factors on population stability

9.2. Attractors and Their Types

When the dynamics of a population model is traced over a large number of generations, then the model exhibits an asymptotic behavior which is almost independent from initial conditions (e.g., initial population density). Asymptotic trajectory is called "attractor" because model trajectories converge to this asymptotic trajectory.

There may be several attractors in a model. In this case each attractor has a domain of attraction. Model trajectory converges to that attractor in which domain initial conditions were located.

In this example, there are two attractors: a limit cycle (at the left) and a stable equilibrium (at the right). Domains of attraction are colored blue, they never overlap. For different starting places (initial conditions), trajectories converge to different attractors.

Types of attractors:

1. Stable equilibrium (=steady state)

2. Limit cycle

3. Chaos

Examples:

Stable MfDHxiun pdffll oo

Ganihuous Discrete EflfiCydè ErilCydè

9.3. Equilibrium: Stable or Unstable?

Equilibrium is a state of a system which does not change.

If the dynamics of a system is described by a differential equation (or a system of differential equations), then equilibria can be estimated by setting a derivative (all derivatives) to zero.

Example: Logistic model

To find equilibria we have to solve the equation: dN/dt =

This equation has two roots: N=0 and N=K. An equilibrium may be stable or unstable. For example, the equilibrium of a pencil standing on its tip is unstable; The equilibrium of a picture on the wall is (usually) stable.

An equilibrium is considered stable (for simplicity we will consider asymptotic stability only) if the system always returns to it after small disturbances. If the system moves away from the equilibrium after small disturbances, then the equilibrium is unstable.

The notion of stability can be applied to other types of attractors (limit cycle, chaos), however, the general definition is more complex than for equilibria. Stability is probably the most important notion in science because it refers to what we call "reality". Everything should be stable to be observable. For example, in quantum mechanics, energy levels are those that are stable because unstable levels cannot be observed.

Now, let's examine stability of 2 equilibria points in the logistic model.

In this figure, population growth rate, dN/dt, is plotted versus population density, N. This is often called a phase-plot of population dynamics. If 0 < N < K, then dN/dt > 0 and thus, population grows (the point in the graph moves to the right). If N < 0 or N > K (of course, N < 0 has no biological sense), then population declines (the point in the graph moves to the left). The arrows show that the equilibrium N=0 is unstable, whereas the equilibrium N=K is stable. From the biological point of view, this means that after small deviation of population numbers from N=0 (e.g., immigration of a small number of organisms), the population never returns back to this equilibrium. Instead, population numbers increase until they reach the stable equilibrium N=K. After any deviation from N=K the population returns back to this stable equilibrium.

The difference between stable and unstable equilibria is in the slope of the line on the phase plot near the equilibrium point. Stable equilibria are characterized by a negative slope (negative feedback) whereas unstable equilibria are characterized by a positive slope (positive feedback).

The second example is the bark beetle model with two stable and two unstable equilibria. Stable equilibria correspond to endemic and epidemic populations. Endemic populations are regulated by the amount of susceptible trees in the forest. Epidemic populations are limited by the total number of trees because mass attack of beetle females may overcome the resistance of any tree.

Endemic Epidemic equilibrium equilibrium

Tnresfuici density winch is sJTlctentto start an outbreak

Stability of models with several variables

Detection of stability in these models is not that simple as in one-variable models. Let's consider a predator-prey model with two variables: (1) density of prey and (2) density of predators. Dynamics of the model is described by the system of 2 differential equations:

This is the 2-variable model in a general form. Here, H is the density of prey, and P is the density of predators. The first step is to find equilibrium densities of prey (H*) and predator (P*). We need to solve a system of equations:

The second step is to linearize the model at the equilibrium point (H = H*, P = P*) by estimating the Jacobian matrix:

Third, eigenvalues of matrix A should be estimated. The number of eigenvalues is equal to the number of state variables. In our case there will be 2 eigenvalues. Eigenvalues are generally complex numbers. If real parts of all eigenvalues are negative, then the equilibrium is stable. If at least one eigenvalue has a positive real part, then the equilibrium is unstable.

Eigenvalues are used here to reduce a 2-dimensional problem to a couple of 1-dimensional problem problems. Eigenvalues have the same meaning as the slope of a line in phase plots. Negative real parts of eigenvalues indicate a negative feedback. It is important that ALL eigenvalues have negative real parts. If one eigenvalue has a positive real part then there is a direction in a 2-dimensional space in which the system will not tend to return back to the equilibrium point.

There are 2 types of stable equilibria in a two-dimensional space: knot and focus

There are 3 types of unstable equilibria in a two-dimensional space: knot, focus, and saddle

Stability in discrete-time models

Consider a discrete-time model (a difference equation) with one state variable:

This model is stable if and only if :

where 1 is the slope of a thick line in graphs below:

You can check this yourself using the following Excel spreadsheet:

Excel spreadsheet "ricker.xls"

If the slope is positive but less than 1, then the system approaches the equilibrium monotonically (left). If the slope is negative and greater than -1, then the system exhibits oscillations because of the "overcompensation" (center). Overcompensation means that the system jumps over the equilibrium point because the negative feedback is too strong. Then it returnes back and again jumps over the equilibrium.

Continuous-time models with 1 variable never exhibit oscillations. In discrete-time models, oscillations are possible even with 1 variable. What causes oscillations is the delay between time steps. Overcompensation is a result of large time steps. If time steps were smaller, then the system would not jump over the equilibrium but will approach to it gradually.

Now we will analyze stability in the Ricker's model. This model is a discrete-time analog of the logistic model:

First, we need to find the equilibrium population density N* by solving the equation:

This equation is obtained by substituting Nt+1 and Nt with the equilibrium population density N* in the initial equation. The roots are: N* = 0 and N* = K.. We are not interested in the first equilibrium (N* = 0) because there is no population. Let's estimate the slope df/dN at the second equilibrium point:

Now we can apply the condition of stability:

Thus, the Ricker's model has a stable equilibrium N* = K if 0 < r < 2.

If a discrete time model has more than one state variable, then the analysis is similar to that in continuous-time models. The first step is to find equilibria. The second step is to linearize the model at the equilibrium state, i.e., to estimate the Jacobian matrix. The third step is to estimate eigenvalues of this matrix. The only difference from continuous models is the condition of stability. Discrete-time models are stable (asymptotically stable) if and only if all eigenvalues lie in the circle with the radius = 1 in the complex plain.

9.4. Quantitative Measures of Stability

In the 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.

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 = W"1 KJMtf]"1 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 * to the variance of log population numbers

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 Ricker's model we can use the linearized model at the equilibrium point:

■ 0 ft Mi U, where N is log population density, and £ is the white noise with a zero mean. Noise is not correlated with log population numbers. Thus:

T

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-

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. Get a reprint! (PDF)

We found that Ricker's model is stable if 0 < r < 2. The question is what happens to model population when stability is lost. The first suggestion is that population will get extinct. But this is the wrong answer. Below there are simulations of population dynamics using Ricker's model with different values of r:

233.

9.5. Limit Cycles and Chaos r = 0.5 Monotonous increase in numbers. K = 200

r = 1.9 Damping oscillations.

In the two upper figures the model has a stable equilibrium, only the patterns of approaching the equilibrium are different. In the three lower figures there is no stable equilibrium. Non-equilibrium dynamics may be of 2 types: a limit cycle when the trajectory repeats itself, and chaotic when the trajectory does not repeat itself.

The bifurcation plot (below) helps to visualize all types of dynamics generated by the Ricker's model. This graph is plotted as follows: for each value of parameter r which is incremented with 0.05 steps, population dynamics was simulated for 200 generations. First 125 generations were discarded because the population may not have reached the asymptotic behavior. Population numbers in the rest 75 generations are plotted versus the value of parameter r:

r < 2 Stable equilibrium r = 2 Bifurcation into a 2-point limit cycle r = 2.5 Bifurcation into a 4-point limit cycle

Then there is a series of cycle duplication: 8 points, 16 points, etc.

For r > 2.7 there are some regions where dynamics returns to a limit cycle, e.g., i=3.15.

Chaotic dynamics looks like stochastic noise, however the model is absolutely deterministic. Chaotic models are widely used for random number generation in computers.

One of the questions that are often discussed in ecological literature is "does chaos really exists in population dynamics?". The major argument in favor of chaos is: when model parameters are fit to known time series of population dynamics then the dynamics of the model with these parameters is chaotic. Another kind of arguments is based on attempts to separate chaotic dynamics from stochastic noise. However detection of chaos is difficult because of several problems:

• There is no evidence that the model is correct. Usually these models ignore many ecological processes (natural enemies, etc.). It is necessary to use multiple models for detecting chaos (Ellner and Turchin 1995, Amer. Natur. 145: 343375).

• The confidence interval for parameter values is usually large enough to cover both chaotic and non-chaotic (limit cycle) model dynamics.

• Time series of population dynamics are usually not long enough to separate chaotic dynamics from stochastic noise.

At this point, chaotic dynamics was detected consistently only in a few microtine populations (Ellner and Turchin 1995). I suspect that in these cases chaotic dynamics was induced by the seasonal cycle in population numbers. Chaos was never detected in time series with 1 year as a time step. Probably chaos is a rare phenomenon in population dynamics.

Questions and Assignments to Lecture 9

9.1. Population dynamics is simulated by the differential equation:

Draw schematically the phase portrait (rate of population growth vs. population density). Find all equilibrium points and characterize their stability.

9.2. Characterize the type of attractor in population dynamics: (a) stable equilibrium; (b) limit cycle, (c) chaos.

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