On Nonergodic Uniform Lotka–Volterra Operators

It is well known that any classical Lotka-Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved.

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“…Note that

V_{normalΓ, boldr}

is an identity mapping if

γ_{1} = γ_{2} = γ_{3} = 0

. We notice that historical behavior of more general LV operators has been investigated in previous works 41–43 …”

Section: Resultsmentioning

confidence: 91%

“…We notice that historical behavior of more general LV operators has been investigated in previous works. [41][42][43] The main result of this section is to establish that V Γ,r is a one-to-one on S 2 . This result is formulated later on.…”

Section: Resultsmentioning

confidence: 99%

A class of bijective Lotka–Volterra operators and its application

Mukhamedov

Hee

Rosli

2023

Math Methods in App Sciences

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“…if the dimension of the simplex is more than 2, then particular cases of

W

have been investigated in previous studies 23,25,34 …”

Section: Resultsmentioning

confidence: 99%

“…(ii) if 𝑓 (x) ≡ Const; g(x) ≡ 0; s(x) ≡ 0; h ≡ 1, then the corresponding W was studied in Ganikhodzhaev and Zanin 31 (iii) if g(x) ≡ 0, s(x) ≡ 0 and 𝑓 (x) is a quadratic polynomial with h ≡ 1, then the corresponding W has been studied in previous studies 24,27,32,33 ; (iv) if s(x) ≡ 0, 𝑓(x), g(x) and h are arbitrary, then the corresponding W has been studied in Saburov 1 ; (v) if the dimension of the simplex is more than 2, then particular cases of W have been investigated in previous studies. 23,25,34 Example 2.3. Assume that 𝑓 (x) ∶ [0, 1] → [0, 1] is an increasing continuous function.…”

Section: Resultsmentioning

confidence: 99%

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Historical behavior for a class of Lotka–Volterra systems

Jamilov

Mukhamedov

2022

Math Methods in App Sciences

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It is well-known that discrete-time Kolmogorov systems play important role in studying predator-prey models with interactions. A topic of this paper deals with stochastic discrete-time Kolmogorov systems which are called Lotka-Volterra operators. In the present paper, we provide a new class of LV operators, and establish that each of operator from this class has historical behavior, that is, their time averages diverge.

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Dynamics of nonlinear stochastic operators and associated Markov measures

Jamilov,

Mukhamedov,

Mukhamedova

et al. 2024

Math Methods in App Sciences

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The aim of this paper is to examine the stability of a class of nonlinear stochastic operators on a finite‐dimensional simplex. We identify a number of properties of the set of the class of stochastic operators, including the stability of each operator. We then use these operators to define non‐homogeneous Markov measures that rely on initial data and carry out an investigation into the absolute continuity and singularity of these measures.

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On Nonergodic Uniform Lotka–Volterra Operators

Cited by 22 publications

References 10 publications

A class of bijective Lotka–Volterra operators and its application

A class of bijective Lotka–Volterra operators and its application

Historical behavior for a class of Lotka–Volterra systems

Dynamics of nonlinear stochastic operators and associated Markov measures

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