solutions assignment 3 – larch bud moth cycles · 2015. 2. 18. · solutions assignment 3...

Solutions Assignment 3 –Larch Bud Moth Cycles Systems Ecology: Principles and Modelling (Fischlin & Lischke) - FS 2015

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Solutions Assignment 3 – Larch Bud Moth Cycles

The underlying model approach was deliberately chosen as simple as possible. This since the essential, namely the approach, comes more clearly to the fore, and in its universality can rather be understood.

Questions and answers

1) Problem questions

What are the causes behind the population cycles of the larch bud moth (Zeiraphera diniana GN.)? To which extent is a predator-prey model suitable to answer this question? What do you conclude from your investigation on the possibility to control larch bud moth as a pest insect?

See the general epistemological basic situation (handouts Fig. I-5) and please put yourself in the position of a systems analyst. This implies that the population system of larch bud moth in the subalpine Alps represents the real system and you find yourself in following situation:

As several pest control experiments have shown (e.g. Auer, 1974), the question whether it is possible to control larch bud moth can only be answered if we understand, which mechanisms regulate the population. We need to know which of the ecological factors causes actually the larch bud moth population to cycle.

2) Starting from these questions sketch a possible way forward to answer them.

If we would succeed in developing a mathematical model than can simulate the real larch bud moth system, this might help us to understand the regulatory mechanisms causing the larch bud

Solutions Assignment 3 –Larch Bud Moth Cycles Systems Ecology: Principles and Modelling (Fischlin & Lischke)

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moth cycles and then to design with the use of the model a possible control strategy. To control the larch bud moth means to find a method to keep the density of larvae below that of the defoliation threshold (timing of application, intensity of treatment, etc.). For instance we could promote the presence of antagonists (biological control) combined with the application of insecticides targeted in such a manner, that it kills mainly larch bud moth larvae, but spares the natural enemies (e.g. microbiological control using BT1).

3) Write down all steps needed to solve the problem while using a known mathematical model (Note: Try to use the classical model of Lotka-Volterra with population cycles).

Follow the usual steps of a systems analysis:

Write down the problem question: What you want is a model that describes the population dynamics of larch bud moth by mimicking the typical periodical fluctuations as observed in the altitudunal zone of optimal development (1600 – 2200 meters above sea level).

Prepare, collect, and sift facts, experimental data, and connections: As done during the class

Sketch a verbal/qualitative model by drawing a relational graph: The system universe X consists of the larch bud moth lbm and an abstract antagonist complex a:

X = { lbm, a }

The system structure R, derived using the 2nd order predicate P(x,y)) ::= "x influences the rate of change of y" yields the following structure set R:

R = { (lbm, lbm), (lbm, a), (a, lbm), (a, a) }

These are the reasons why the 2nd order predicate for each of the listed ordered pairs of the structure set R becomes true: The Larch bud moth as well as the antagonist grow only in function of themselves (e.g. once eradicated, they can no longer grow and their rate of reproduction depends on their numbers) => (lbm, lbm) and (a, a). The more larch bud moth individuals are present, the better the antagonist can reproduce => (lbm, a). The antagonist influences larch bud moth negatively by killing its host => (a, lbm). In the following relational graph the system is represented to be equivalent to R:

Figure 1: Relational graph of the autonomous "predator-prey" system formed from the system elements (state variables) larch bud moth (lbm) and antagonist (a). The system structure R can also be written as a set of ordered pairs R = { (lbm, lbm), (lbm, a), (a, lbm), (a, a) }. = system border.

1 Bacillus thuringiensis (microbial pest control)


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The system has no inputs and outputs and is thus autonomous, i.e. independent of its environment. Any influences by weather or humans are neglected.

Building the mathematical model:

The classical Lotka-Volterra model for a predator-prey relationship seems to be suitable:

€

˙ x 1 = a*x1 - b*x1*x2 (1)

€

˙ x 2 = b'* x1* x2 - c* x2

y1 = x1 y2 = x2 (2)

whereas x1 prey one has: a, b, b', c > 0 x2 predator

The continuous time system of equation (1) consists of 2 coupled, non-linear differential equations (DESS). It generates periodic population cycles, as can be observed in the real system of larch bud moth. In addition, there is evidence that some of the more than 100 antagonist species of larch bud moth also fluctuate periodically and can cause quantitatively significant mortality in the larch bud moth population. Hence we can infer:

Symbol Significance Unit System-

theoretical significance

x1 Larval density of larch bud moth #/kg LB2 SV3 x2 Antagonist density, understood as representative of

an abstract antagonist-complex, which consists mainly of parasitoids as Eulophidae (e.g. Elachertus argissa WALKER, Sympiesis punctifrons THOMPSON, Dicladocerus westwoodi WESTWOOD) and Ichneumonidae (e.g. Phytodietus griseanae KERRICH)

#/kg LB SV

a Intrinsic relative growth rate of larch bud moth /a MP4 b Infestation rate kg LB/#/a MP b' Intrinsic relative growth rate of antagonists due to

consumption of arch bud moth larvae. b' ≈ 0.1*b kg LB/#/a MP

c Relative mortality rate of the antagonists /a MP x1(0) Initial value of the density of larch bud moth larvae #/kg LB IV5 x2(0) Initial value of the antagonist’s density #/kg LB IV

2 larch branches (fresh weight) 3 state variable 4 model parameter 5 initial value


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To use the model it is necessary to find first the specific values for all parameters. This means it is necessary to fit the classical Lotka-Volterra model as close as possible to the observations from the larch bud moth system. This is done first in steps 5 "Calibration", 7 "Finding initial and/or boundary conditions, domains, and simulation parameters" and 10 "Model and parameter identification". Once all this has beend one are we ready to tackle the problem question by designing a controller such that at least in the model the density of the larch bud moth larvae can be kept below that of the defoliation threshold of about 100 larvae per kg LB. This is, however, not required for this assignment, but will be covered by the lecture.

Calibration: The calibration is skipped in its strict sense by assuming that x1 corresponds exactly to the density of larch bud moth larvae and x2 to that of the antagonist (see Output equations (2)).

4) Examine now the behavior of your selected model according to the steps described under 3) (Notes: Use Easy ModelWorks for the simulation of the model. Try first to estimate roughly the missing or poorly known model parameters. Once you have a rough estimate try to identify these parameters more precisely, by running many simulations until the behavior of the simulation model matches as close as possible the figure shown below. Use a population density of 0,018 larvae / kg larch branches (value from figure, see below) as an initial value for the larch bud moth population in 1949. Further use following values for the model parameters as rough first estimates: intrinsic growth rate of the prey population and mortality rate of the predator population, respectively: ≈1.0/a; predation rate: ≈0.5/a/predator; assume for the biomass transfer from one trophic level to the next higher one a loss of about 90%). Remember that the observed data (1949-1986) are as follows (linear scale):

Figure 2: Observed densities of larvae of larch bud moth Zeiraphera diniana GN. in the Upper Engadin Valley. Densities are shown as larvae per kg larch branches (fresh weight) (#/kg LB).

Build the simulation model (Implementation of the mathematical model in form of a simulation model): For our model (1) it makes sense to create a simulation model, because the system of not linear, coupled differential equations is analytically not easy to solve. For this purpose the modelling- and simulation tool "Easy Model Works" is used. We need then only to formulate the model equations (1), which correspond to a DESS6 according to the syntax of EMSL3 «Easy Model Works».

6 Acronym for Differential Equation System Specification


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“Easy ModelWorks” provides already a predefined Lotka-Volterra model, which we can use to simulate the population growth of larch bud moth. This ´sample model´ can be opened by the menu command Modelling > Sample Models > Lotka-Volterra. In this predefined form, however, it is not usable for the larch bud moth and needs some adjustments: First, we have to delete the self-inhibition term of the first differential equation, which determines the population dynamics of larch bud moth by choosing the menu command Modelling > Edit model... (Figure 3).

Figure 3: The self-inhibition term is deleted from the pre-defined prey equation of the sample model "Lotka-Volterra" in the dialogue box by choosing the menu command Modelling > Edit model...

The model equations (1) hold only four instead of five parameters. Thus we need to reduce the number of model parameters by choosing the menu command Modelling > Change Model type (Figure 4).

Figure 4: Dialogue to change the number of model parameters by choosing the menu command Modelling > Change Model type. In our example the number of model parameters is reduced from 5 to 4.

To simulate the population dynamics of larch bud moth with the Lotka-Volterra model we need first to determine the model parameters a, b, b' and c in (1) specific for larch bud moth. The parameters a, b, and c are given in the assignment: a is the intrinsic growth rate of the prey population = 1.0 per year, b is the predation rate = 0.5 per year and per density of antagonists, i.e. per antagonist per kg LB, and c is the mortality rate of the antagonist = 1.0. b' is the intrinsic relative growth rate of the antagonist regarding the consumption of larch bud moth larvae. It is assumed that 90% of the biomass is lost during the flow from one trophic level to the next. So b' = 0.1 b (10% of b) = 0.05.


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Math. Symbol

Identifier of the model parameters used in the simulation model

Range Unit

t t [t0.. t*] a7

�

˙ x 1 x1Dot #/kg LB/a

�

˙ x 2 X2Dot #/kg LB/a

x1(0) x1.(0) 0.018 #/kg LB

x2(0) x2.(0) 2 #/kg LB

a a 1.0 /a

b b 0.5 kg LB/#/a

b' b2 0.5*0.1=0.05

kg LB/#/a

c c 1.0 /a

Choosing the menu command Modeling > Edit Model... the identifiers (names) of the model parameters as well as the name of the simulation model itself can be altered (Figure 5).

Figure 5: Choosing the menu command Modeling > Edit Model... the identifier of the model parameters and the name of the simulation model can be modified. Be aware that the identifiers of the model parameters as used in the differential equations must match exactly those used in the fields below the equations where the values are assigned to the model parameters.

Finding initial and/or boundary conditions, domains, and simulation parameters:

The initial value of state variable x1 is given with 0.018 larvae/kg LB. If we arbitrarily assume that the initial growth rate of the larch bud moth population is 0, we can determine the missing initial value of state variable x2 by inserting all parameter values in the first differential equation (1) as follows:

7 a – year (annus)


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�

˙ x 1(0) = 0 = 1.0*x1(0) – 0.5*x1(0)*x2(0)

0 = 1.0*0.018 – 0.5*0.018*x2(0)

x2(0) = 2

Both initial values are now determined (see also Figure 5).

We can insert the start and the end points of the simulation time, say 1949 and 1986, and specify the maximum values of x1 and x2 so that the shape of the curves become visible. This is done by the menu command Simulation > Set time... (Figure 6) and Simulation > Set monitoring... (Figure 7), respectively.

Figure 6: Dialogue to modify the start and end points of the simulation time by choosing the menu command Simulation > Set time...

Figure 7: Dialogue to adjust the maximum values (max) for the plotting of the state variables x1 and x2 (x1 and x2) by choosing the menu command Simulation > Set monitoring....

Simulation (conducting experiments with the simulation model):

With all of the above specified parameters the results of the simulation are as shown in Figure 8:


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Figure 8: Initial simulation results obtained with the larch bud moth model using the Lotka-Volterra model.

5) What can you learn from your simulation experiments? When and why did you encounter particular difficulties? How fit is the model to help you answer the problem questions? What can be inferred from the results for a possible control strategy?

Interpretation of simulation results:

By comparing the behavior of the Lotka-Volterra model (Figure 8) with the observations (Figure 2) we see that the simulated and the observed data do not match well. For example, the simulated period length is too big (13 a instead of 8.9 a). Also, the amplitude is a bit too small (200 instead of 260 larvae/kg LB). With the chosen model parameters the Lokta-Volterra model is obviously not able to reproduce the real larch bud moth system in a realistic manner. We can try to modify the parameters so that the model behaviour fits the periodic cycles of the larch bud moth population better. That means we will try a "manual" parameter identification in the next step.

Model and parameter identification:

If we believe the model to be still useful, we can assume that larch bud moth is in fact more strongly controlled by the antagonist than we have assumed so far (see step 7 „Finding initial and/or boundary conditions, domains, and simulation parameters“). We can therefore try to increase the effect of the antagonist by increasing their density. Perhaps this results in a more realistic model behaviour. We can do that by increasing the value of the antagonist population (x2) to 20 by choosing the menu command Modelling > Edit model... (Figure 9):


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Figure 9: Attempt to increase the effect of the antagonist by increasing its initial value x2(0) to 20 (for resulting model behaviour see Figure 10).

This attempt results in the following simulation (Figure 10):

Figure 10: Results from the simulation experiment that attempts to increase the control effect of the antagonist by increasing its initial value x2(0) to 20 (see also Figure 9): The length l of the population cycles is prolonged.

The antagonist and larch bud moth achieve a higher density. Periodic cycles still arise, but they are now less frequent.

We recognize that the increase of the initial value of the antagonist caused a prolongation of the cycles (the length l becomes with x2(0) = 20 => l = 20 a; with x2(0) = 10 => l = 15 a). However, the average cycle of the larch bud moth population as estimated from several decades of research is 8.9 a. Therefore we need to have shorter cycles. Since an increase resulted in longer cycles, let us try the opposite and run the model with a lower population of the antagonist (Figure 11 and 12).


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Figure 11: Simulation experiment to reduce the period length l by reducing the initial value x2(0) of the antagonist to 0.01 animals per kg larch branches (see Figure 12).

Figure 12: Simulation experiment to reduce the period length l by reducing the initial value x2(0) to 0.01 antagonists per kg larch branches (see Figure 11).

We get some improvement but this modification gives us not yet to the desired outcome, because the period length l of the population cycles is still too long. Obviously, we have to continue adjusting model parameters. Perhaps the larch bud moth population could reach its maximum value faster by increasing a, the intrinsic, relative growth rate, and the population cycles might then become shorter. By increasing the parameter a from 1 to 3 we obtain the following simulation results (Figure 13):


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Figure 13: Simulation results when the model parameter a is increased from 1 to 3 per a. The aim was to shorten the period length l.

As Figure 13 shows, the simulated model behaviour is now indeed more similar to the observed behaviour (Figure 2 (real larch bud moth system)) than this was the case with all previous parameter constellations. But we have overdone it, the outbreaks are now too close together. We could try to decrease the relative mortality rate of the antagonist c slightly. . Perhaps a decrease of the model parameter c would cause the individual cycles to get longer and outbreaks would then occur less often. A reduction of c (from 1 to 0.8) produces the following result (Figure 14):

Figure 14: Simulation experiment attempting to extend the period length l (compare with Figure 13). This was indeed achieved by decreasing the model parameter c from 1 to 0.8.


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Model validation:

The solution shown in Figure 14, with a period length of the population cycles of approximately 9 years, fits the real population dynamics of the larch bud moth population quite well (Figure 2). These are the initial conditions and parameter values used:

Element Value

x1(0) 0.018

x2(0) 0.01

a 3.0

b 0.5

b' 0.5*0.1 = 0.05

c 0.8

Note that above simulation results (Figure 14) can be only reproduced exactly if the parameters for the numerical integration are set exactly like this: integration method Runge-Kutta 4th order, integration step h = 0.1/a, monitoring interval hm = 0.1/a. The model is numerically very sensitive due to the stability behaviour of the used Lotka-Volterra model (cf. Figure 16).

Model application:

Based on the previous investigations we can draw the following conclusions:

• We could finally achieve a quite acceptable agreement between the model behaviour and the observations.

• If indeed the model mimics accurately the actual behaviour of the real system, it would be easy to control larch bud moth populations: A one-time application of an insecticide that reduces the density of larch bud moth to approximately 15 larvae/kg LB would be sufficient, given this happens at a time when the predator density is about 6 predators/kg LB (Figure 15). From then on, the damage threshold of 100 caterpillars per kg LB would never be reached again, let alone exceeded.

Figure 15: Population dynamics of larch bud moth according to the Lotka-Volterra predator-prey model (for parameter values see Table under Step ) after a successful application of an insecticide, assuming the population of larch bud moth was reduced to about 15 larvae/kg LB, at a time when about 6 antagonists/kg LB were present. Cycles disappear almost entirely and the defoliation threshold of 100 larvae/kg LB is never reached again (only lower part of the graph shown).


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The question is whether the model, thanks to its good match with the observed data, can now really be used for the application. Unfortunately, this is to be negated. These areor the reasons:

• Agreement between model behaviour and observations or measurements is a necessary condition, but is not sufficient for a useful model application.

• The fit (Figure 14) was found only for one state variable, i.e. larch bud moth, but not necessarily for the antagonists. To achieve this, we would need to have more data, i.e. densities of antagonists, available. Comparison with such observations (see lecture) shows indeed some mismatch between simulated and observed values, diminishing the good first impression we may have gotten from Figure 14.

• There are other key features of the model behavior, in particular the stability behaviour, that is in complete dissent with observations (compare large scale experiment in Goms applying insecticides (AUER, 1974, lecture) with model behavior shown in Figure 15). A full explanation for this difference can be derived from Figure 16:

Figure 16: Phase portrait of the Lotka-Volterra model (version population cycles, see equations (1)), which shows that this model is neutrally stable instead of asymptotically stable. The real larch bud moth system has exhibited asymptotic stability in pest control experiments applying insecticides such as DDT and Dimecron (a phosphamidon8) (Auer, 1974).

Considering all insights gathered thus far, including those detailed in the course, and despite of the found agreement between model and observations (Figure 14) we come to the following main conclusion:

The Lotka-Volterra model is not suitable to answer the problem question and cannot be used to design a control strategy for larch bud moth as a pest insect.

Literature Cited: Auer, C., 1974. Ein Feldversuch zur gezielten Veränderung zyklischer Insektenpopulationsbewegungen.

Schweiz. Z. Forstwesen, 125: 333-358. 8 Organophosphate insecticide: (EIZ)-[3-chloro-4-(diethylamino)-4-oxobut-2-en-2yl] dimethyl phosphate

solutions assignment 3 – larch bud moth cycles · 2015. 2. 18. · solutions assignment 3...

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