# Passivity of Lotka–Volterra and quasi-polynomial systems

## Summary (3 min read)

### 1. Introduction

- Lotka–Volterra systems are widely-used models to describe the dynamic behavior of interactive species or agents [1].
- In the paper [4] the equivalences among the different stability types of Lotka–Volterra systems were discussed.
- This advantageous property is a basis of several results in the field, see e.g. [5–7].
- Passivity theory offers a possibility to overcome this difficulty, that is one of the motivations of their present work.
- It allows a system categorisation in terms of energy transfer between the system and its environment.

### 2.1. Passive systems

- The input-affine system (1) is passive w.r.t.
- If the system (1) is zero state detectable and passive w.r.t.
- Assume that the system (1) is passive w.r.t.

### 2.2. Lotka–Volterra systems

- The notion of Volterra–Lyapunov stability (see [15]) differs from the common Lyapunov stability definition invoked in theorem 2.
- The Volterra–Lyapunov matrix stability is used here to conclude on the asymptotic stability of Lotka–Volterra systems. .
- The strict inequality (8) is a sufficient asymptotic stability condition for the autonomous Lotka–Volterra system.
- If the coefficient matrix M is rank-deficient (i.e. not invertible), this condition cannot be applied.

### 3.1. Passivity of Lotka–Volterra systems

- For passivity analysis define the error state: e = x − x∗, (16) where x∗ is a strictly positive equilibrium point satisfying (5).
- −Ky asymptotically stabilises the equilibrium state x∗.
- It was exploited that the sum of a negative definite matrix and a negative semidefinite matrix having the same dimensions is a negative definite matrix.
- Note, that the full rank of matrix M is not necessary for passivity.

### 3.2. Special Lotka–Volterra systems

- As follows, the passivity properties of two important classes of Lotka–Volterra systems are analysed.
- The coefficients m12, m21 ∈ R>0 determine the inter-influence of the predator–prey population.

### 3.3. Passivity of quasi-polynomial systems

- The generalisation of the passivity-related results for Lotka–Volterra systems to the QP case is based on the joint storage function family of the two system classes.
- The passivity of the QP system (30) can be investigated in the monimial space, i.e. through the Lotka–Volterra variables, by analysing the Lotka–Volterra model that corresponds to (30).
- The error state of the QP model is then defined using the Lotka–Volterra states [i.e. the QP quasimonomials of the QP system (30)].
- This way, the passivity of the QP model can be handled using the above results on the passivity of Lotka–Volterra models and on the fact, that QP and Lotka–Voltera models share their storage functions (6) and (14).

### 4. Feedback equivalence of Lotka–Volterra systems to passive systems

- In the case of a Lotka–Volterra system that is not passive or it is passive only for some set of its parameters, one can use the notion of feedback equivalence to ensure and investigate the passivity.
- The notion of feedback equivalence introduces a class of input-affine systems, that are not necessarily passive, but they can be made passive by applying suitable state feedback to them.
- In the general case, the skew-symmetry property of the interconnection matrix of the controlled system can be assured by choosing u = Kpe + up, Kp ∈ Rm×m. (33) With this input the interconnection matrix (35) Accordingly, the Lotka–Volterra system is equivalent to a passive system where the input is defined in equations (33) and (35).
- This method includes extra parameters into the Lotka–Volterra coefficient matrix M in a bilinear way and turns the LMI (32) into a bilinear matrix inequality, which belongs to an NP-complete problem class.
- This method is not necessary in the present case.

### 5. Disturbance attenuation

- Passivity theory opens the possibility to design powerful controllers not only for stabilising but also for disturbance rejection purposes [20].
- This section proposes a physically meaningful yet realisable controller for disturbance attenuation of Lotka–Volterra systems.

### 5.1. The control problem

- In realistic Lotka–Volterra systems the natural rate cannot be considered constant, its nominal value may change in time.
- The deviation of the rate vector from its nominal value can be viewed as an additive, non-constant, bounded disturbance.

### 5.3. Disturbance attenuation

- Let the prescribed attenuation gain be γ.
- It yields that, if the controller gain matrix is chosen such that k > 1 2 ( 1 + 1 γ ) , (47) then the disturbance attenuation control objective is achieved.

### 6.1. Lotka–Volterra model

- A three-dimensional Lotka–Volterra predator–prey system is considered which describes the behavior of three coexisting species: two predators and one prey.
- The equilibrium states of the predators’ populations always satisfy m12x∗2 + m13x∗3 = l1.
- Without control, the extinction of one of the predator populations can be observed.
- The simulation results presented in figures 3 and 4 show that the proposed disturbance attenuation approach ensures the convergence of the controlled states to the setpoint.

### 6.2. Quasi-polynomial model with m = n

- The examined QP system belongs to the same equivalence class as the Lotka–Volterra model (51) of the previous example.
- For the sake of simplicity, the numerical parameter values of section 6.1 has been used here.
- This also means, that they are dynamically similar.
- These quasimonomials correspond to the Lotka–Volterra system (51)’s state variables x1, x2 and x3, respectively.
- The simulation results presented in figures 7 and 8 show that the proposed disturbance attenuation approach ensures the convergence of the controlled states to the setpoint.

### 6.3. Quasi-polynomial model with m = n

- Usually, the Lotka–Volterra and the QP systems has state spaces of different dimensions [7].
- The typical situation is when the QP system is embedded into the Lotka–Volterra form.
- The Lotka–Volterra embedding inflates the originally n dimensional state space to an m dimensional Lotka–Volterra dynamics.
- The simulation results presented in figures 11 and 12 show that the proposed disturbance attenuation approach ensures the convergence of the controlled states to the setpoint which was determined from the Lotka–Volterra case using the quasi-monomial transformation corresponding to the QP system.
- Similarly to the Lotka–Volterra case, smaller prescribed disturbance attenuation level ensures smaller steady state error.

### 7. Discussion and conclusions

- Using the classical logarithmic Lyapunov function of autonomous Lotka–Volterra and QP systems the authors derive passivity conditions of the open version of these systems.
- This means that one considers the possibility of manipulating the death/birth rate vectors of the species as inputs.
- The condition of passivity in (21) is less strict than the classical one by requiring only semidefiniteness.
- By using the input-extended model of the system, it was shown that the asymptotic stability i.e. the convergence into the equilibrium point, can be achieved by using diagonal state feedback having arbitrarily small gains.
- Moreover, passivity theory provides a novel and powerful approach to design linear static feedback control laws for Lotka–Volterra systems.

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###### Q2. What are the future works mentioned in the paper "Passivity of lotka–volterra and quasi-polynomial systems" ?

It is important to note that one should extend the classical Lotka–Volterra and QP system models with suitable input and output variables for which passivity properties hold. This means that one considers the possibility of manipulating the death/birth rate vectors of the species as inputs. Asymptotic stability of the closedloop system, i. e. the convergence into the equilibrium point, can also be achieved by extending the previous control with diagonal small gain state feedback. By using the input-extended model of the system, it was shown that the asymptotic stability i. e. the convergence into the equilibrium point, can be achieved by using diagonal state feedback having arbitrarily small gains.