paramecium didiniumjgrant3/cw/mcmfinal.pdf#...

A Review of Gary W. Harrison’s “Comparing Predator-Prey Models to Luckinbill’s Experiment with Didinium and Paramecium”

by Jason Grant, Bethany Herila, and Shant Mahserejian

Final Project for ACMS 50730: Mathematical /Computational Modeling Fall 2012 – Dr. Alber

Introduction Mathematical models describing predator-‐prey interaction often base their theory on the Lotka-‐Volterra equations. They provide a good framework for describing the changes in populations between two groups, where one group consumes the other, such as wolves and sheep, lions and antelope, or even on smaller scales dealing with micro-‐bacteria. However, these mathematical equations often don’t provide the detailed circumstances that different predator-‐prey combinations could have in real-‐world circumstances. This was the case for Leo Luckinbill’s experiments in 1973, when he studied the behavior of two groups: Paramecium, and its natural predator Didinium. Due to their rapid growth and consumption rates, the changes in population dynamics of bacteria provide a convenient setting for studying predator-‐prey interplay. Although, from a mathematical point of view, there was more to be understood in order to accurately describe how the populations of these two groups changed when they were put together. Gary Harrison was interested in finding out which governing equations played a dominating role in characterizing the behavior of these two bacterial groups. This paper follows Harrison’s work as he searches for mathematical models that make a better fit to the data collected from Luckinbill’s experiments. 1. The Basic Predator-Prey Model Lotka-‐Volterra’s Model The Lotka (1925) and Volterra (1931) equations provide a general model to predator-‐prey interactions. This basic model defines two rates, and population densities (number per cubic centimeter) over time for the prey and predatory,

€

x(t) and

€

y(t) respectively.

€

dxdt

= ρ 1− xK

⎛

⎝ ⎜

⎞

⎠ ⎟ x −ωf (x)y

dxdt

=σf (x)y − γy

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

where ωf (x) =ωxφ + x

The parameters in this model are characterized as follows:

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ρ is the specific growth rate of the prey;

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K is the carrying capacity of the prey in the absence of predators;

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γ is the rate at which predators die in the absence of prey;

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ω is the maximum rate of prey consumed by a single predator;

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σω is the efficiency of

converting prey to predator;

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φ is the half-‐saturation constant; and

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ωf (x) , the rate at which prey are consumed per predator, also known as the functional response of the predator.

This initial model is quite fundamental in nature. It is assumed that the predator's growth rate is strictly dependent and proportional to the rate of prey consumption. The preliminary extension to the model begins with the functional response of the predator,

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ωf (x) . Evidence from Holling (1959) and Murdock and Oaten (1975) suggest that predators often exhibit a saturation effect in their functional rate when prey is abundant, and gave rise to the Holling Type 2 functional response formula. Parameter dependence In order to display the importance of these parameters in the behavior of the population dynamics simulated by this model, the ODE45 solver in MATLAB was used to solve this system of equations for a given set of non-‐zero parameters. The resulting simulation demonstrates the oscillatory behavior of a predator-‐prey scenario and their stabilizing behavior as time goes on in Figure 1. This behavior is a typical characteristic of the Lotka-‐Volterra Model, and therefore a favored choice for describing a scenario exhibiting this sort of phenomena. Changing the values of each parameter one by one to a non-‐significant value of 0.1 demonstrates how this characterization is lost, as displayed in Figure 2.

Figure 1: A simulation generated by solving the Lotka-Volterra Model using a set of non-zero parameters

Figure 2: A simulation generated by solving the Lotka-Volterra Model, but setting each parameter to the insignificant value of 0.1

one by one is the following order: (top row, left to right)

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γ ,

€

K ,

€

φ , and (bottom row, left to right)

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ρ ,

€

σ ,

€

ω .

Due to the drastic changes in the profiles of the simulated population densities for both predator and prey, it is clear why each parameter serves an important role in creating the characteristic behavior that the Lotka-‐Volterra Model is known for. So, in studying to improve this model, merely changing the parameter values isn’t enough to have it fit observed experimental data. Rather, other considerations and alterations must be made to better describe the governing forces behind a certain real world scenario. Stability Steady-‐state equilibrium of the predator-‐prey model occurs when oscillations between the species' density have dampened and equilibrium values are reached by both prey,

€

x* , and predator,

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y* . The equilibrium point

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E1 = (x*,y*) may be considered stable or unstable within the system, and it is

found by setting

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dxdt

=dydt

= 0, and then solving the equations for

€

x and

€

y . Doing so gives the following

points for stability:

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Prey isocline for dxdt

= 0 : x* = 0 and y* =ρω

1− xK

⎛

⎝ ⎜

⎞

⎠ ⎟ φ + x( ) which is a concave - down parabola

Predator isocline for dydt

= 0 : x* =γφ

σ − γ and y* = 0 which is a vertical line

Using the graphical stability condition of Rosenzwieg and MacArthur,

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E1 is stable if the vertical predator isocline is to the right of the vertex of the parabolic prey isocline, or:

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γφσ −γ

>K −φ2

If the previous inequality is not satisfied, then

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E1 is unstable and oscillations will spiral out toward a stable limit cycle. Analysis shows that the further

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E1 begins to the left of the vertex of the prey

isocline, the faster the oscillations will diverge. Whether the predator and prey coexist depends on how close the oscillations come to the boundaries

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x = 0 and y = 0 . Theoretically, any real-‐value above zero implies the presence of predators or prey; however, in a realistic scenario, a population density below 1 is considered to be extinct. Luckinbill's Experiments Leo Luckinbill conducted an experiment in 1973, which consisted of three predator-‐prey interactive models. Paramecium aurelia was grown together with its predator, Didinium nasatum in 6 mL of standard Cerophyl medium. In the first run, experiment A, the Didinium consumed all the Paramecium within a few hours. During the second phase, expriment B, the medium was thickened using Methyl Cellulose, and the populations went threw two to three cycles before the predator group became extinct. The primary effect of adding Methyl Cellulose was to reduce the swimming speed of both species, although it turned out to have a greater effect on the “hunting ability” of the predator. In the third and final experiment C, the medium was changed to half-‐strength Cerophyl, but also thickened with Methyl Cellulose. By doing so, the carrying capacity of the Paramecium was reduced and again allowed the Didinium to move at as slower rate. Luckinbill showed that reducing the nutrient supply for the paramecium by half reduced only decreased the growth rate slightly while mostly decreasing the carrying capacity from approximately 900 individuals/mL to 400 individuals/mL. This resulted in the amplitude of the stable limit cycle decreasing to a smaller radius, and produced oscillating populations, which were sustained over several cycles (33 days) without either population going extinct. Though the final experiment lasted for 33 days, a constant population density was never reached. Qualitatively, it is suggested that by further thickening the medium and/or using a weaker Cerophyl medium, a stable system could be produced with the population settling to constant density values. Luckinbill's experiments showed that the basic predator-‐prey model is a good start for a qualitative match, however, it performed poorly in computing the numerical results to make a qualitative match. Therefore, finding a mathematical model that accurately described Luckinbill’s experiments was desired. 2. Error Estimates, Solution Method, and Relevant Values In order to quantify values pertaining to the mathematical model, an error estimation system is established to provide a measure for comparing various types of models, but also to find appropriate values for parameters and initial conditions that would fit the real observed data values. As for the experimental data, it is reported that the numbers for experiments A & B where only available in the form of graphs from Luckinbill’s report, while the numbers for experiment C were provided by Luckinbill directly via personal communication. So, in order to retrieve values for experiments A & B, digital analysis was applied to their respective graphs, and therefore they suffered an additional source of error. Furthermore, the real data from experiments A & C proved to be difficult to fit, so therefore the preliminary error was first computed against the data from experiment B. Once a match was found for this set of data, the necessary parameters were adjusted appropriate to make comparisons for the data sets form experiments A & C. Methods and Error Criteria The Marquardt-‐Levenberg method was implemented to estimate the parameters in the differential equations, which were then solved using a variable step Runge-‐Kutta method. These simulated results were matched to the experimental data in the least squared sense in order to find the best fit. This meant parameter values were chosen based on minimizing the error term S2 defined by:

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S2 =x(ti) − xobserved (ti)

sxi

⎡

⎣ ⎢

⎤

⎦ ⎥

2

+i=1

N

∑ y(ti) − yobserved (ti)syi

⎡

⎣ ⎢

⎤

⎦ ⎥

2

i=1

N

∑

The terms

€

x(ti) and

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y(ti) are the simulated values for prey and predator densities, respectively, at time steps

€

ti . The terms

€

xobserved (ti) and

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yobserved (ti) are values from experimental data. The terms

€

sxi and

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syi are weights representing the relative measurement errors in the observation. Though several options for these weights were considered, the best fitting model showed no difference between utilizing a weight, versus calculating unweighted error values. Also, other error values were calculated using a fixed prey growth rate, and altering intial conditions, but they did not pose much of a difference in the error analysis, especial in the case of the best fitting model. For the sake of simplicity, the unweighted error values (where

€

sxi = syi =1) while allowing movement in all of the parameters, represented by S12, is the only one focused on for this report. Initial Conditions Luckinbill reported that the initial number of organisms at the start of his experiments:

€

x(0) =15 Paramecium/mL and y(0) = 5.833 Didinium/mL . However, it is also reported that the first 0.5 day of measurements do not reflect the predator-‐prey interactions predicted by the suggested mathematical models, because as admitted by Luckinbill, “well fed Didinium” were placed in the experimental medium. The error value computed by taking this time delay into consideration did not differ significantly from the S12 value for the best fitting model; therefore, it is not discussed in more detail in this report. Parameter Values Along with the experimental data for the cases described by experiments A, B, and C, measurements of Paramecium grown alone were taken for both regular and weaker solutions of Cerophyl. This data was used to compute the prey’s carrying capacity in both cases as K=898 and 400 respectively. The rest of the parameters (for each respective model) were found by numerically finding values that provided a best fit in the least squares sense to the real data from experiment B. In addition, it was reported that a case was never observed where two very different sets of parameters both giving good fits to the observed data. Therefore, it seemed practical to only estimate parameters to a degree of accuracy no higher than 5%. After all, the goal of the study was not to determine parameters, but what types of behavior take place in this predator-‐prey interaction. 3. Finding a Quantitative Match I. Basic Lotka-‐Volterra Model: When using the basic model utilizing reasonably sized parameters, it predicts the correct number of oscillations in the population densities, hence providing a good qualitative match. Though this sets a solid place to start, the quantitative fit was described as “disappointing”. The resulting error value was S12=236,137. Due to this large error, there was great interest to alter the model with possible considerations that would improve the error, and match the conditions theoretically describing the governing equations of the experiment more closely. II. Predator Mutual Interference: At first glance, the most obvious mismatch is that the predators seem to grow too rapidly. This led to attempting to introduce competition amongst the predators for space or some resource other than the prey. This was done by including a new function

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g(y) into the model as follows:

€

dxdt

= ρ 1− xK

⎛

⎝ ⎜

⎞

⎠ ⎟ x −ωf (x)g(y)

dydt

=σf (x)g(y) − γy

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪


and g(y) =y

1− βy

Here,

€

β > 0 is the rate of reduction in predators at high predator densities, hence controlling the population of predators when too many of them came to be. This alteration in the model produced simulations that returned an error of S12=120,313, which is about 50% of the basic model error, a tremendous improvement for a first attempt. Although, as far as the numbers were concerned, the predator density’s peaks occurred too soon, and the population levels dropped too low to correctly represent the experimental data. III. Ratio-‐Dependent Functional Response: This consideration was based on the fact that the predator functional response does not depend on the prey density alone, but on the ratio of prey to predators

€

xy . So, the functional response term was redefined and gave us the following model:

€

dxdt

= ρ 1− xK

⎛

⎝ ⎜

⎞

⎠ ⎟ x −ωf (x,y)y

dydt

=σf (x,y)y − γy

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

where ωf (x,y) =ω( xy )φ + ( xy )

Unfortunately, using this ratio-‐dependent functional response based model was far too stable to represent Luckinbill’s experiments, especially for the set of parameter values chosen. Therefore, this type of correction was abandoned. IV. Leslie-‐type model: In order to control the predator population, an attempt was made to introduce a carrying capacity for the predator that is proportional to the prey density. This required redefining the differential equation for the predator’s population rate of change.

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dxdt

= ρ 1− xK

⎛

⎝ ⎜

⎞

⎠ ⎟ x −ωf (x)y

dydt

=σ 1− γyx

⎛

⎝ ⎜

⎞

⎠ ⎟ y

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪


Making this change to the basic model resulted in an error of S12=143,491, about 60% of basic model error. Note that this is not as good as the improvement by considering predator mutual interference. Furthermore, the Leslie-‐type model still contained systematic errors, including incorrectly predicting the addition of Methyl Cellulose to the medium. This consideration was also abandoned.

V. Sigmoid functional response: So far, all previous variations predicted that the prey densities drop to values much lower than the real data measurements. This suggests that the predator reduces its searching rate when prey populations become scarce. The suggested correction was to apply a sigmoid functional response to predator density by redefining

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f (x) in the model defined by the predator mutual interference.

€

dxdt

= ρ 1− xK

⎛

⎝ ⎜

⎞

⎠ ⎟ x −ωf (x)g(y)

dydt

=σf (x)g(y) − γy

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

where f (x) =x

φ + x1− 1+νx( )e−νx[ ] and g(y) =

y1− βy

The resulting error for this model configuration is S12=93,546 (40% of basic model error) by using a rate of predator reduction of

€

β = 0.00057 , which is only slightly better when the rate was set to

€

β = 0. Hence, including the predator mutual interference effect is unnecessary for its added parameter and complexity. Also, the visual fit is not as good as the predator mutual interference alone, in particular the predator densities. It makes the predation rates too low and drives prey to extinction, and therefore leaving more room for improvement. VI. Delayed Predator numerical response: Another offset made by previous versions of the model predicted that the predator density respond instantaneously with changes in prey densities. They showed it to be quicker than what the real data suggests, resulting in peaks in the density graphs occurring too early. To fix this issue, a time lag applied in the predator’s numerical response by assuming (1) prey consumption creates an inflow of energy stored, and (2) overall predator growth rate is an increasing function of average energy per predator. So, a new variable z is introduced representing the total energy storage of predators. The model is rescaled for simplification and combined with a sigmoid functional response to be stated as follows:

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dxdt

= ρ 1− xK

⎛

⎝ ⎜

⎞

⎠ ⎟ x −ωf (x)y

dzdt

=σf (x)y −δz

dydt

= z − γy

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

where f (x) =x

φ + x1− 1+νx( )e−νx[ ]

Here the term

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δz represents the predator’s reproductive rate. This configuration of the model gives the best fit to the experimental data with an error value of S12=30,084, only 13% of basic model error, which is a tremendous improvement. Though other combinations using the standard functional response and the predator mutual interference were attempted, they did not yield the optimal results of this version of delayed predator response with the sigmoid functional response. Continuing in this manner by introducing a delay in prey growth only made slight improvements, hence not justifying the complexity of the involvement of additional parameters associated with it.

4. Conclusion Since the standard Lotka-‐Volterra model utilizing Holling’s functional response equation only provided a qualitative match the results from Luckinbill’s experiments, several alterations were considered in an attempt to find a quantitative fit to the observed data. The optimal model to describe the predator-‐prey recorded interplay was based on the criteria not only for the best fit in the least squares sense, but also for simplicity as to avoid the complexity introduced by minimally beneficial parameters. Therefore, using a Delayed Predator Numerical Response along with a Sigmoid Functional Response was decided to be the best model to describe the population dynamics between Paramecium aurelia and its natural predator Didinium nasatum. The resulting simulation compared with the real data is shown in Figure 3.

Figure 3: Best fit of the model with Delayed Predator Response and Sigmoid Functional Response for (solid) vs. observed data from Luckinbill’s experiments (dashed) for experiment B.

The model parameters numerically determined for the best fit of this version of the model are as follows:

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K = 898 ρ = 3.02 ω = 9.74φ = 54.3 σ = 9.15 γ =1.78ν = 0.0983 δ =1.78 z(0) = 90.45

Reporting on the success of this study, the researchers made a significant improvement from the standard Lotka-‐Volterra predator-‐prey model, to the final version by eliminating 87% of the error in the least squares fit to Luckinbill’s observed experimental data. By including the delaying the response of a predator consuming prey, and converting it to energy for reproduction, as well as redefining the functional response to a Sigmoid type, the real world scenario was described more accurately. As successful as their efforts were, there were some shortcomings in this study that could have improved the rigor and trueness of the numbers represented. First, attention is turned to the data values used from experiments A & B. The physical printed graphs from the 1973 study published by Luckinbill were actually used via digital analysis, as opposed to having the actual measured values representing the true data. This digitization in itself introduced errors that were not discussed, and could have been avoided had the actual numbers been used, as in the case for experiment C. The discrepancy in the data values, though probably small, could have effected the values in sharp peaks appearing in Luckinbill’s graphs, and therefore compromising the error estimates used when fitting the model.

Second, there was some lack of confidence in the values of some parameters that were chosen. In particular, the prey’s growth rate,

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ρ , was left variable while fitting with the S12 error, whereas it was fixed to the value of 3.3 when calculating the S22 error. The reasons for this decision were left vague, though a greater emphasis was put onto the S12 value through the research. Finally, no details on how population measurements were reported. Thus, the order of accuracy in the population measuring process was not declared, leaving the “goodness” of the measured data unclear. This was considered an important fact to skip, because the searching for a better fitting model used error analysis criteria, which ended in the S12=30,000 range. An open question of “Why did we stop here?” is not addressed, leaving uncertainty about the best fitting model that was settled upon. Perhaps there are other considerations that would have improved the mode even further, up to the level of accuracy of the errors incurred during the population measurement process. Providing more information from this angle would have strengthened the argument that the final model chosen was indeed the best model to decribe Luckinbill’s experiments. As discussed by the author, there is also more room for continuing the study to the next level. It is a good idea to repeat the experiment to retrieve better data using measurements with more frequent time increments. The current data used time steps of 0.5 days. The initial conditions were also questioned, since well-‐fed Didinium was used, and therefore delaying the anticipated predator-‐prey interaction. This new set of experiments could also use an attempt to achieve the stable conditions claimed by finding the right Cerophyl and Methyl Cellulose combination as predicted by the stability analysis. Of course, this research made a final claim that there is a best fitting model to the Paramecium vs. Didinium scenario, which is begging for new data to conduct verification. Reference Harrison, G. W. 1995. Comparing Predator-‐Prey Models to Luckinbill's Experiment with Didinium and Paramecium. Ecology 76: 357-‐374.

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