# Discrete Competitive Lotka–Volterra Model with Controllable Phase Volume

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Competitive Lotka–Volterra Model

_{i}represents the growth rate of species i and a

_{ij}the competitive effect species j has on i as they compete for resources. The competitive Lotka–Volterra system is based on the logistic population model, and K

_{i}represents the carrying capacity of species i.

_{i}—the growth rate of the number of partnerships

_{n}—partnerships of various types

#### 2.2. Finite-Difference Models with Controllable Symmetry

#### 2.2.1. Semi-Implicit Integration as a Tool to Obtain Adaptive Discrete Systems

_{D}

_{1}and Φ

_{D}

_{2}[31], one can obtain the symmetric composition CD method of order 2

#### 2.2.2. Discrete CLVM Model with Controllable Symmetry

_{i}is the growth rate of number of partnerships, x

_{n}—number of partnerships of various types, n—the number of existing types of partnerships. As was previously mentioned, the direct calculation in Equation (6) can be replaced by the simple iterations algorithm [32], which results in the so-called semi-explicit (SED) method.

## 3. Results

#### 3.1. Phase Space Analysis

_{i}was set to 1 in all of the experiments.

#### 3.2. Bifurcation Analysis

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Phase space of proposed competitive Lotka–Volterra model with controllable symmetry (CSCLV) model with S = 2; (

**b**) Phase space of proposed CSCLV model with S = 1; (

**c**) Phase space of proposed CSCLV model with S = 1.5; (

**d**) Phase space of proposed CSCLV model with S = 1.1. The red line shows the attractor of the competitive Lotka–Volterra system with no phase volume control (“real volume”).

**Figure 2.**Phase volume changes of CD-based CSCLV model while symmetry is varied as S = cos (wt). The red line shows the phase space volume of CSCVLM with fixed symmetry S = 0.5.

**Figure 3.**Phase-volume diagram of SED CSCLVM with the cosine function applied as symmetry law. The red line shows initial phase volume for a system with fixed symmetry S = 0.5.

**Figure 5.**Bifurcation diagram for SED-based CSCLVM. The system is chaotic in the whole range of the symmetry parameter S.

**Figure 6.**Bifurcation diagram for CD-based CSCLVM. The system is chaotic for all values of the symmetry parameter S.

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Voroshilova, A.; Wafubwa, J.
Discrete Competitive Lotka–Volterra Model with Controllable Phase Volume. *Systems* **2020**, *8*, 17.
https://doi.org/10.3390/systems8020017

**AMA Style**

Voroshilova A, Wafubwa J.
Discrete Competitive Lotka–Volterra Model with Controllable Phase Volume. *Systems*. 2020; 8(2):17.
https://doi.org/10.3390/systems8020017

**Chicago/Turabian Style**

Voroshilova, Anzhelika, and Jeff Wafubwa.
2020. "Discrete Competitive Lotka–Volterra Model with Controllable Phase Volume" *Systems* 8, no. 2: 17.
https://doi.org/10.3390/systems8020017