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MBW:Observability of Trophic Food Chains

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Executive Summary

In the paper "Observer Design for Open and Closed Trophic Chains" by Z. Varga, M. Gamez, and I. Lopez the authors explore mathematical techniques for observing systems from partial information. A simple model of a tropic food chain is created, containing three species. Under the assumption that only one of these species can be observed, the authors demonstrate the circumstances under which it is possible reconstruct the state of all species in the model.

Overview

Mathematics Used

The authors for this paper used dynamic modeling and the mathematics behind control theory. Along with this, Jacobian's were used to find steady states of the tropic states.

Types of Model

This paper uses a non-Lotka Volterra type trophic system which means that it does not simply follow a predator-prey relationship, but rather is multi leveled and contains open and closed chains. For a discussion on Lotka Volterra type population system see: to The Dynamics of Arthropod Predator-Prey Systems

Biological System Studied

The biological system studied was a Trophic chain that involved three levels. The 0th level in this chain was the "resource", the 1st tropic level was the "producer" and the 3rd trophic level was the "primary consumer". In the closed form, dead individuals from the 1st and 2nd tropic level could be recycled back into the 0th tropic level where as in the open form there is no recycling.

Context

The motivating example is ecology, where it may be practical to monitor only a small subset of the ecosystem. Their ecological model was a trophic chain. A trophic chain is a food chain where there is a linear food chain, where each species feeds on the previous. All species die and replenish some portion of a resource, which sustains the creature at the bottom. In the author's model, there is a resource, producer and primary consumer. For example, nutrients, plants, and herbivores would fit this model. If an ecologist can construct a mathematical model of how the system behaves, physical observations may give insight into other parts of the system. The mathematics behind this procedure are borrowed from the field of control theory. An extension is if a system can be observed, it can also be controlled, although modifying the system based on observation goes beyond the scope of this paper.

For another study on a trophic system with two competing predators see: Three Species Food Chain Modeling

History

There have been a number of publications on the application of observer design and control theory to ecology. An example is a previous paper, by Lopez, Gamez and Molnar called "Observability and observers in a food web", where they analyzed a similar problem. They looked at a simple food web. There have been application beyond ecology also by Varga, Gamez and Lopez called "Observer design for phenotypic observations of genetic processes".

Model

The model is based on three elements of a trophic chain, x_{{0}},x_{{1}},x_{{2}}. These represent the nutrient, producer, and consumer.

{\begin{array}{c}{\dot  x}_{{0}}=Q-\alpha _{{0}}x_{{0}}x_{{1}}+\beta _{{1}}m_{{1}}x_{{1}}+\beta _{{2}}m_{{2}}x_{{2}}\\{\dot  x}_{{1}}=x_{{1}}(-m_{{1}}+k_{{1}}\alpha _{{0}}x_{{0}}-\alpha _{{1}}x_{{2}})\\{\dot  x}_{{2}}=x_{{2}}(-m_{{2}}+k_{{2}}\alpha _{{1}}x_{{1}})\end{array}}


The definition of these parameters are:

m_{{1}},m_{{2}} - The malthus parameters. These force exponential decay in the absence of food.

\alpha _{{0}}x_{{0}},\alpha _{{1}}x_{{1}} - rate at which species 1 consumes 0, rate species 2 consumers 1.

k_{{1}},k_{{2}} - rate at which species 1, 2 increase biomass from their consumption

\beta _{{1}},\beta _{{2}} - rate at which species 1, species 2 die and recycle to nutrient


This set of equations is also referred to as f({\bar  x}). All parameters are greater than one, but k_{{1}},k_{{2}},\beta _{{1}},\beta _{{2}} are less than 1.

Analysis

The authors define two matrices:.

A=f'(x*),C=h'(x*)

Where A is a system of n equations. C is the observation, which is found from the definition of h(x)=x_{{i}}-x_{{i}}* where i is a species in the chain. For their model, they chose

C=(1,0,0)

which observes the resource. The asterisk denotes a non-trivial steady state. This can be found in a tedious but straightforward manner.

They also define a matrix H which maps the observation to a matrix of the same dimensions as A.

H=\left({\begin{array}{c}h1\\0\\1\end{array}}\right)

This also assumes that h_{{1}}>max({\frac  {m_{1}x_{1}*}{m_{2}x_{2}*}},{\frac  {\alpha _{0}x_{0}*}{\beta _{2}k_{2}}}).

A system can be reconstructed from observations of a single system if

(1)~~rank[C|CA|CA^{2}|...|CA^{{n-1}}]^{T}=n

assuming the observation is near equilibrium. This is tedious result from control theory, that the authors use without proving.

A-HC must also have negative and real eigenvalues. The authors use the criteria that the characteristic polynomial should satisfy the Hurwitz criteria.

The matrix A, after linearization of the system is:

A=\left({\begin{array}{ccc}-\alpha _{0}x_{1}*&-\alpha _{0}x_{0}*+\beta _{1}m_{1}&\beta _{2}m_{2}\\k_{1}\alpha _{0}x_{1}*&0&-\alpha _{1}x_{1}*\\0&k_{2}\alpha _{1}x_{2}*&0\end{array}}\right)

The authors then find the equilibrium state and linearize the system There are solutions if:

Q>{\frac  {m_{{1}}m_{{2}}}{\alpha _{{1}}k_{{1}}k_{{2}}}}-{\frac  {\beta _{{1}}m_{{1}}m_{{2}}}{\alpha _{{1}}k_{{2}}}}

This simply says that if there are too few resources, the food chain may not have a stable steady state.

They then state a result from control theory. They can construct a local exponential observer:

(2)~~{\dot  z}=f(z)+H[y-h(z)]

where y=h(x). A way to think about this is that an observation near equilibrium (mapped to the system by H) will exponentially decay to the true state.

The authors proceed to graph the trajectories, however they graph them as a phase plot. I have graphed the solutions of the first trajectory, as a function of time. It is clear that even though the trajectories have slightly different conditions, all populations converge to the same steady state.

Sim1.jpg

Sim2.jpg

Sim3.jpg

The authors then use the same procedure to demonstrate trajectories based on observation of all three species are possible.

Useful Math

There is a useful mathematical technique to place the eigenvalues of a matrix of the for A-BK where A and B are known, and K is to be determined to give the desired eigenvalues.

It is possible to write out all the eigenvalue equations of a system by writing:

(A-BK){\bar  w}={\bar  \lambda }{\bar  w}

Instead of individual eigenvalues {\bar  w} is a vector of all eigenvectors w_{1},w_{2},...w_{n}, and {\bar  \lambda } is a matrix with corresponding eigenvalues on the diagonals. This can be rearranged as

A{\bar  w}-BK{\bar  w}-{\bar  \lambda }{\bar  w}=0

Which can be rewritten as a block matrix equation:

\left({\begin{array}{cc}A-{\bar  \lambda }&-B\end{array}}\right)\left({\begin{array}{cc}{\bar  w}\\K{\bar  w}\end{array}}\right)=0

Re-write again by replacing the vector by z:

\left({\begin{array}{cc}A-{\bar  \lambda }&-B\end{array}}\right)z=0

The vector z is in the nullspace of the matrix. It can be written as z_{{bot}}=Kz_{{top}}. This allows a solution for K, as K=z_{{bot}}(z_{{top}})^{{-1}}. By multiplying the vector out to obtain the equation (A-{\bar  \lambda })z_{{top}}=Bz_{{bot}} it is possible to solve for K, and select whatever eigenvalues you want:

K=B^{{-1}}(A-{\bar  \lambda })

Connection to Control Theory

The authors do not explicitly explain the connection to control theory or the motivation for equation (2). The first step is to write out the linearized system:

{\dot  x}=Ax,y=Cx

Then write out another system, that is an estimation of the system:

{\dot  {{\hat  {x}}}}=A{\hat  x}+H(y-{\hat  y}),{\hat  y}=C{\hat  y}

In system the H matrix is multiplied by the difference between the true and estimated observation of some species. Subtracting system from results in:

{\dot  x}-{\dot  {{\hat  x}}}=A(x-{\hat  x})-H(y-{\hat  y})

The difference in x can be renamed as z, and the result is the local observer.

{\dot  z}=Az-H(y-{\hat  y})

Substituting in the linearized definition of y, the system becomes:

{\begin{array}{cc}{\dot  z}&=Az-HC(x-{\hat  x})\\&=Az-HCz\\&=(A-HC)z\end{array}}

It would be great if asymptotically decayed to the true system. It is for this reason it is desired that A-HC has negative real eigenvalues. Pole placement can be used to achieve this if desired.

I will briefly comment on what this has to do with control theory, as Robustness Analysis of an Observer-Based Controller in a Food Web gives a good description of control theory. A system you want to control can be described as

{\dot  x}=Ax+BKx

where A is the linearized model of the system. B is known, and K is chosen to make the system behave as desired. The criteria (1) will determine whether the system is controllable. This is the same criteria that determines if a system is observable.

Conclusion

The authors have demonstrated that given complete information about one species, as well as a mathematical model of the interactions between the species, it is possible to determine the state of all the species in the system, under certain criteria. Using this in practice may be more difficult. Even if the model is known, determining the steady state may be challenging. The author also does not comment on over how quickly the system asymptotically decays to the true solution.

Discussion of Recent Citation

The paper entitled Monitoring in a predator-prey systems via a class of high order observer design cites this paper and is by Juan Luis Mata-Machuca, Rafael Martinez-Guerra, and Ricardo Aguilar cites this paper in particular for it's use of high order observation in order to construct a model of a predator-prey relationship. The paper explains that observer design allows for a system to be modelled in various different fields and gives examples such as population genetics, evolutionary dynamics and neural-networks. This paper uses a high order polynomial to understand a predator-prey relationship. The paper used various resources such as theoretical model as well as numerical results. Then this methodology was used on other various Lotka Volterra models with up to three species with success.

External links

The paper: Observer design for open and closed trophic chains