In the classical Lotka-Volterra population models, the interacting species affect each other's growth rate. We propose an alternative model, in which the species affect each other through the limitation coefficients, rather then through the growth rates. This appears to be more realistic: the presence of foxes is not likely to diminish the fertility of rabbits, but will contribute to limiting rabbit's population. Both the cases of predation and of competition are considered, as well as competition in case of periodic coefficients. Our model becomes linear when one switches to the reciprocals of the variables. In another direction we use a similar idea to derive a multiplicity result for a class of periodic equations.

#### Keywords

- Explosive growth
- predator-prey
- competing species

#### MSC 2010

- 34C11
- 34C25
- 92D25

One way of solving the logistic population equation (here ^{2}, and obtain a linear equation for
^{2}, the second one by ^{2}, and setting

Let us compare (1.2) with the classical Lotka-Volterra predator-prey model

Similarly to the Lotka-Volterra model, the proposed model (1.2) predicts oscillatory behavior for predator-prey interaction, and either stable coexistence or competitive exclusion for competing species. Unlike the Lotka-Volterra model, it is possible that the population number of one of the species goes to infinity in finite time, while the number of the other species remains finite and positive. Explosive growth of populations occurs often in nature, in connection with various “outbreaks”, see e.g., D. Ludwig, D.D. Jones and C.S. Holling [4] for a model of spruce budworm. Notice that our analysis leads to some non-standard questions about linear systems. For example, if a solution of (1.3) starts in the first quadrant of the

Using the Floquet theory, we analyze a case of predator-prey interaction with periodic coefficients, and give a condition for the existence of a limit cycle.

In another direction we derive a multiplicity result on the periodic solutions for a class of periodic equations

Consider the model
^{2},

The system (2.1) has a rest point (_{1} and _{2} are determined from the initial values (

_{1}_{2}_{t→T}_{t→T}_{t→T}_{t→T}

The change of variables
_{t→T}_{t→T}_{t→T}_{t→T}

It remains to prove the lower bounds for the periodic solutions in the first part of the theorem. From (2.4) one sees that the positivity of

Using

Consider the model

We begin with a simple observation: if

Setting
_{0},_{0}) given by
_{0},_{0}) lies in the first quadrant if either
_{0} and _{0}, we translate the rest point to the origin, obtaining the system
_{0},_{0}) is a stable node, while in case (3.5) holds, one eigenvalue is negative and the other one is positive, so that (_{0},_{0}) is a saddle.

_{t→T}_{t→T}

The change of variables
_{1} and _{2} are the eigenvalues of _{1} and _{2} are given by (3.8), while _{0} and _{0} are given by (3.3).

(i) In case (3.5) holds, the eigenvalues of _{1} < 0 < _{2}. The term _{1}^{λ1t}_{1} is negligible in the long run. The eigenvector _{2} corresponding to the positive eigenvalue (“plus” in front of the square root) has one component positive, and the other one is negative. It follows that all of the solutions of (3.2) eventually move either northwest or southeast of the rest point (_{0},_{0}) intersecting either the

(ii) In case (3.4) holds, the general solution of (3.2) is given by (3.9), with negative _{1} and _{2}. It follows that the point (_{0} > 0,_{0} > 0) as

In case (3.4) holds, the solution of (3.2) connects the points (_{0},_{0}) and (_{0},_{0}) and (_{t→T}

We now consider a periodic perturbation of the explosive predator-prey model
_{1} and _{2} of _{0}(

_{p}_{p}

Observe that the Floquet multipliers of (4.4) satisfy _{i}_{1} = 1, then from the first line in (4.5)_{2} = 1, giving a contradiction in the second line in (4.5), because
_{p}_{p}_{p}_{p}

We derive next an a priori bound on _{p}_{p}_{0}, for some constant _{0}. Indeed, integrating both equations in (4.2) over (0,_{1} and _{2}. The desired bound over (0,

We claim that _{p}_{p}_{1} and _{2}, and
_{0}(_{1} = _{2} = 0. The vector _{0}(_{p}_{p}

The transformation
_{i}_{0}(

What if one changes the _{0}(^{3} term to _{0}(^{2n+1}? In case it is _{0}(^{5}, the method of V.A. Pliss [6] still gives the same result with a little extra effort. For higher powers things get more involved, and in fact existence of at most three

This theorem follows from a more general result of A. Sandqvist and K.M. Andersen [7]. They considered the equation (5.2) on the interval (0,

A simpler proof of the Theorem 5.1 was found in P. Korman and T. Ouyang [3]. We now simplify the presentation in that paper. The proof will follow from the following three simple lemmas.

Calculate _{x}^{2}_{xxx}

The proof is standard, and it can be found in e.g., P. Korman [2], p. 245. The next lemma is crucial.

Set

Turning to the proof of the Theorem 5.1, observe that different solutions of (5.2) do not intersect by the uniqueness theorem. If the equation (5.2) has four _{zzz}

Equations of the type (5.2) occur often in ecological problems, see e.g., S. Ahmad and A.C. Lazer [1], or P. Korman [2].

_{xn}_{xn}

Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Nonlinear Mathematical Modelling of Bone Damage and Remodelling Behaviour in Human Femur Value Creation of Real Estate Company Spin-off Property Service Company Listing Entrepreneur's Passion and Entrepreneurial Opportunity Identification: A Moderated Mediation Effect Model Applications of the extended rational sine-cosine and sinh-cosh techniques to some nonlinear complex models arising in mathematical physics Study on the Classification of Forestry Infrastructure from the Perspective of Supply Based on the Classical Quartering Method A Modified Iterative Method for Solving Nonlinear Functional Equation A comprehensive evaluation of county economies in the Beijing-Tianjin-Hebei Region based on entropy TOPSIS analysis New Principles of Non-Linear Integral Inequalities on Time Scales Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Analysis of the agglomeration of Chinese manufacturing industries and its effect on economic growth in different regions after entering the new normal Study on the social impact Assessment of Primary Land Development: Empirical Analysis of Public Opinion Survey on New Town Development in Pinggu District of Beijing Possible Relations between Brightest Central Galaxies and Their Host Galaxies Clusters and Groups Attitude control for the rigid spacecraft with the improved extended state observer An empirical investigation of physical literacy-based adolescent health promotion MHD 3-dimensional nanofluid flow induced by a power-law stretching sheet with thermal radiation, heat and mass fluxes The research of power allocation algorithm with lower computational complexity for non-orthogonal multiple access The art design of industrialised manufacturing furniture products based on the simulation of mathematical curves Research on the normalisation method of logging curves: taking XJ Oilfield as an example A Method of Directly Defining the inverse Mapping for a HIV infection of CD4+ T-cells model On the interaction of species capable of explosive growth Research on Evaluation of Intercultural Competence of Civil Aviation College Students Based on Language Operator Combustion stability control of gasoline compression ignition (GCI) under low-load conditions: A review Research on the Psychological Distribution Delay of Artificial Neural Network Based on the Analysis of Differential Equation by Inequality Expansion and Contraction Method The Comprehensive Diagnostic Method Combining Rough Sets and Evidence Theory Study on Establishment and Improvement Strategy of Aviation Equipment Design of software-defined network experimental teaching scheme based on virtualised Environment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode System dynamics model of output of ball mill The Model of Sugar Metabolism and Exercise Energy Expenditure Based on Fractional Linear Regression Equation Constructing Artistic Surface Modeling Design Based on Nonlinear Over-limit Interpolation Equation Statistical analysis of typical elevator accidents in China from 2002 to 2019 Optimal allocation of microgrid using a differential multi-agent multi-objective evolution algorithm About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function Numerical Simulation Analysis Mathematics of Fluid Mechanics for Semiconductor Circuit Breaker Dynamics of infectious diseases: A review of the main biological aspects and their mathematical translation Cartesian space robot manipulator clamping movement in ROS simulation and experiment Effects of internal/external EGR and combustion phase on gasoline compression ignition at low-load condition Research of urban waterfront space planning and design based on children-friendly idea Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Human Body Movement Coupling Model in Physical Education Class in the Educational Mathematical Equation of Reasonable Exercise Course Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Impact of Web Page House Listing Cues on Internet Rental Research on management and control strategy of E-bikes based on attribute reduction method A study of aerial courtyard of super high-rise building based on optimisation of space structure Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation