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}

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