# Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models

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## Abstract

**:**

## 1. Introduction

## 2. Single Population Dynamics in a Model with $\mathbf{\delta}$-Shaped Growth Rates

## 3. Paradoxes in a $\mathit{\delta}$-System of Two Competitors

- 1.
- A low-productive population can displace a highly productive one. Put ${\tau}_{1}=0$, ${\tau}_{2}=2.99$, ${\mu}_{1}=3$, ${\mu}_{2}=2.9$. From (13), we get ${y}_{1}^{\ast}<0$, ${y}_{2}^{\ast}>0$.

- 2.
- The displacement can be non-transitive. Consider a set of three populations with parameters$${\tau}_{1}=1/2,\phantom{\rule{1.em}{0ex}}{\tau}_{2}=3/2,\phantom{\rule{1.em}{0ex}}{\tau}_{3}=5/2,\phantom{\rule{2.em}{0ex}}{\mu}_{1}={\mu}_{2}={\mu}_{3}=6.$$

- 3.
- Coexistence can be non-transitive. Denote by ${x}_{i}$~${x}_{j}$ the coexistence relation. It appears that this relation need not be transitive. For$${\tau}_{1}=0,\phantom{\rule{1.em}{0ex}}{\tau}_{2}=4/3,\phantom{\rule{1.em}{0ex}}{\tau}_{3}=8/3,\phantom{\rule{2.em}{0ex}}{\mu}_{1}={\mu}_{2}={\mu}_{3}=6$$

## 4. A Sufficient Condition for Competitive Displacement—Universal Stock Constant

**Theorem**

**1**

**Theorem**

**2**

**Theorem**

**3**

**.**Assume that the stock condition

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Property**

**A1.**

**Proof.**

**Property**

**A2.**

**Proof.**

## Appendix B

## Appendix C

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**Figure 2.**Competition outcomes for different locations of isoclines: (

**a**) competitive displacement, (

**b**) sustainable coexistence.

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**MDPI and ACS Style**

Il’ichev, V.G.; Rokhlin, D.B.
Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models. *Mathematics* **2024**, *12*, 125.
https://doi.org/10.3390/math12010125

**AMA Style**

Il’ichev VG, Rokhlin DB.
Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models. *Mathematics*. 2024; 12(1):125.
https://doi.org/10.3390/math12010125

**Chicago/Turabian Style**

Il’ichev, Vitaly G., and Dmitry B. Rokhlin.
2024. "Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models" *Mathematics* 12, no. 1: 125.
https://doi.org/10.3390/math12010125