POSITIVE PERIODIC SOLUTION FOR GENERALIZED BASENER-ROSS MODEL

. This paper is devoted to the existence of at least one positive periodic solution for generalized Basener-Ross model with time-dependent co- eﬃcients. Our proof is based on Man´asevich-Mawhin continuation theorem, Leray-Schauder alternative principle, ﬁxed point theorem in cones. Moreover, we obtain that there are at least two positive periodic solutions for this model.

1. Introduction. The Easter Island, which once flourished, and finally, due to the rapid population growth and excessive use of resources, the people on the island was almost extinct. According to records, Basener-Ross [2] in 2004 developed a model of resources and population on an isolated island as follows where x and y represent the total amount of resources and the population of the island, respectively. k is the capacity of the island to carry resources, and c is the resource growth rate when the resources on the island are far less than the carrying capacity, a is the net growth rate of the population when resources are abundant, hy represent the amount of resources that human acquired, and h is the harvesting constant. It is obvious that a, c, h, k are positive constants. According to model (1), residents will survive rely on the existing resources on the island [11,13] . If resources are used excessively, humans will increase rapidly in a short time and eventually disappear. If resources are used reasonably, people and resources will always coexist. That is to say, the rate of human access to resources is not always greater than the rate of growth of resources, otherwise there will be a catastrophe of human. In other words, h ≥ c does not always stand up.
Of course, the Basener-Ross model is just an ideal model. In real life, both changes in the external environment and human living habits will lead to changes in parameters in the model. For example, the changes of climate conditions, the occurrence of natural disasters, the development and evolution of granite landforms, the changes of surface temperatures, and changes in the proportion of elements in air and soil all will lead to changes in resource carrying capacity and resource growth rate (see [3], [1], [8], [7]). In this paper, we consider the case that the parameters c, k are functions about t. Then system (1) is transformed into the following form where a, h are positive constants and c, k ∈ C(R, (0, +∞)) are ω-periodic functions. It is meaningful to study the existence of positive ω-periodic solutions for the Basener-Ross model (2). From the point of view of scientific development, it is closed relationship between human development and resource utilization, as well as the important conditions for sustainable development.
Substituting the above equation into the first equation of system (2), we get the following second order differential equation y (t) + p(t)y (t) + q(t)y(t) = α (y (t)) 2 y(t) + β(t)y 2 (t), where p(t) = a + c(t) − 2h, q(t) = (h − c(t))a, α = 2 − h a , β(t) = − ac(t) k(t) . It is clear that the derivable term is relatively complex, in order to get around the derivable term, the change of variable y = u µ , where µ = 1 1−α . Due to definitions of µ and α, it is easy to see that µ = 0 and µ = −1. With the change, converting equation (3) into the following form It is readily seen that the existence of positive ω-periodic solutions to model (2) reduces to the existence of positive ω-periodic solutions to equation (4). The rest of this paper is organized as follows. In section 2, applying Manásevich-Mawhin continuation theorem, we prove that equation (4) has at least one positive ω-periodic solution, which can be used the cases of a strong singularity and without singularity. In section 3, using Green function and Leray-Schauder alternative principle, we get second existence results for equation (4), the results are applicable to the case of a strong singularity, as well as the case of a weak singularity. In section 4, fixed point theorem in cones is applied to present third existence results for equation (4). Moreover, we prove that equation (4) has at least two positive ω-periodic solutions.

Existence results (I).
In this section, we investigate the existence of a positive ω-periodic solution for equation (4) by using Manásevich-Mawhin continuation theorem. At first, embed equation (4) into the following equation family with a parameter λ ∈ (0, 1]: By applications of Theorem 3.1 in [9], we give the following lemma which will be used to prove the existence of a positive ω-periodic solution for equation (4).
Lemma 2.1. Assume that there exist constants E 1 , E 2 , E 3 with E 2 , E 3 > 0 and E 1 < E 2 such that the following conditions hold: (1) Each possible ω-periodic solution u(t) to equation (5) such that E 1 < u(t) < E 2 for all t ∈ [0, ω] and u < E 3 , here u := max (2) Each possible solution C to equation Then equation (4) has at least one ω-periodic solution.
Next, we investigate the existence of a positive ω-periodic solution for equation (4) by applying Lemma 2.1. According to the value ranges of µ, we get two existence results of positive ω-periodic solutions for equation (4) without a singularity (i.e. −1 < µ < 0) and with a strong singularity (i.e. µ ≤ −2).
Proof. Firstly, we claim that the set of all ω-periodic solutions for equation (5) are bounded. Let be an arbitrary ω-periodic solution of equation (5). We claim that there is a point where D := max{|D 1 |, D 2 } and D 1 , D 2 are defined by condition (H 1 ).
Hence, from equations (5), (10), (17), (18) and (19), the following inequality is given that Let Then condition (1) of Lemma 2.1 is satisfied. For possible solution C to equation Therefore, condition (2) of Lemma 2.1 holds. Finally, we consider the condition (3) of Lemma 2.1 is also satisfied. In fact, from condition (H 1 ), we can observe that So condition (3) of Lemma 2.1 is also satisfied. By application of Lemma 2.1, we get that equation (4) has at least one ω-periodic solution.
In the following, we prove that ω-periodic solution for equation (4) must be positive ω-periodic solution. From equation (8) and the bounds for µ, we get (3) and the relation between y and u, it is clear that which implies ω-periodic solution for equation (4) must be positive ω-periodic solution.
From Theorems 2.2 and 2.3, we only consider the existence of a positive ωperiodic solution for equation (4) in the cases that −1 < µ < 0 and µ < −2 by applications of Manásevich-Mawhin continuation theorem. Therefore, we need to find other methods to consider the existence of a positive ω-periodic solution for equation (4) in the cases that µ > 0 and −2 < µ < −1.

3.
Existence results (II). In this section, we investigate the existence of a positive ω-periodic solution for equation (4) in the cases that µ > 0, −1 < µ < 0 and µ < −1 by using Leray-Schauder alternative principle. At first, we discuss positivity and negativity of Green function of the linear equation corresponding to equation (4).
Consider the following nonhomogeneous second order linear differential equation where b ∈ C(R, R) is an ω-periodic function. Then equation (34) have unique ω-periodic solution which can be written as where G(t, s) is the Green function of equation (34) [4]. Throughout these sections 3 and 4, assume one of the following conditions is satisfied: ( In the following, we study conditions (F 1 ) and (F 2 ). Let X with norm u := max t∈[0,ω] |u(t)| and X be a Banach space, where X is defined by equation (6). On the basis of [12], Cheng and Ren [5] in 2018 proved the Green function is positive for all (t, s) ∈ [0, ω] × [0, ω] if the following condition is satisfied: (A 1 ) There are continuous ω-periodic functions a 1 (t) and a 2 (t) such that ω 0 a 1 (t)dt > 0, ω 0 a 2 (t)dt > 0 and Applying the methods of [12] and [5], we get that the Green function is negative for all (t, s) ∈ [0, ω] × [0, ω] if one of the following conditions is satisfied: (A 2 ) There are continuous ω-periodic functions a 3 (t) and a 4 (t) such that (A 3 ) There are continuous ω-periodic functions a 5 (t) and a 6 (t) such that ω 0 a 5 (t)dt < 0, ω 0 a 6 (t)dt > 0 and a 5 (t) + a 6 (t) = p(t), a 5 (t) + a 5 (t)a 6 (t) = q(t) µ , for t ∈ R.
Next, we state and prove the existence of a positive ω-periodic solution for equation (4). Our proofs are based on the following Leray-Schauder alternative principle.
Lemma 3.1. (see [6]) Assume B 1 is a relatively compact subset of a convex set K in a normed space X, and 0 ∈ B 1 . Let T : B 1 → K be a compact map, then one of the following two conclusions holds: (I) T has at least one fixed point in B 1 .
In 1997, O'Regan improved the existing result by applying Leray-Schauder alternative principle to singular differential equations, namely the following lemma. Lemma 3.2. (see [10]) Assume B 2 is a relatively compact subset of a convex set K in a normed space X, and ρ ∈ B 2 . Let T : B 2 → K be a compact map, then one of the following two conclusions holds: (I) T has at least one fixed point in B 2 .
It is clear that 0 < l ≤ L if condition (F 1 ) holds; l ≤ L < 0 if condition (F 2 ) holds. And 0 < δ i ≤ 1, i = 1, 2. Define set It is easy to verify that K i is a cone in X.

Proof. Consider the family of equations
where σ 1 ∈ (0, 1). An ω-periodic solution of equation (36) is just a fixed point of the operator equation where T 1 is a continuous operator defined by Let B 1 1 := {u ∈ X : u < r 1 }, we claim that T 1 (B 1 1 ) ⊂ K 2 . For any u ∈ B 1 1 , from equation (35), we arrive at which shows that T 1 (B 1 1 ) ⊂ K 2 . In addition, from the continuity of G(t, s) and µ u µ+1 (t), we deduce that T 1 : B 1 1 → K 2 is a continuous and completely continuous operator.
Proof. Similar to the proof procedure of Theorem 3.3, we consider the family of equations where σ 2 ∈ (0, 1). An ω-periodic solution of equation (39) is just a fixed point of the following operator equation where the expression of T 2 is the same as T 1 . Let B 2 1 := {u ∈ X : u < r 2 }, from equation (35), for any u ∈ B which shows that T 2 (B 2 1 ) ⊂ K 1 and T 2 : B 2 1 → K 1 is a continuous and completely continuous operator.
Remark 5. If −1 < µ < 0 and µ > 0, the nonlinear term β(t) µ u µ+1 (t) has not singular, we discuss the existence of a positive ω-periodic solution for equation (4) by using Lemma 3.1. However, if µ < −1, equation (4) has a singularity of repulsive type at the origin, we verify the existence of a positive ω-periodic solution for equation (4) by applications of Lemma 3.2.
Proof. Let f (t, u(t)) = β(t) µ u µ+1 (t). And consider the family of equations whereσ ∈ (0, 1), and n . An ω-periodic solution of equation (42) is just a fixed point of the operator equation where ρ = 1 n and T n is a continuous operator defined by Let B 2 := {u ∈ X : u < r 3 }, we claim that T n (B 2 ) ⊂ K 1 . For any u ∈ B 2 , from equation (41), we arrive at From the equation above, it is clear that where we used the fact 0 < δ 1 ≤ 1. This shows that T n (B 2 ) ⊂ K 1 . In addition, from the continuity of G(t, s) and f , we deduce that T n : B 2 → K 1 is a continuous and completely continuous operator.
We claim that u(t) > 1 n . In fact, from u =σ T n u + (1 −σ)ρ =σ ω 0 G(t, s)f n (s, u(s))ds + 1 n and ω 0 G(t, s)f n (s, u(s))ds > 0, we can see u(t) > 1 n holds. Note that From condition (H 6 ), we can choose n 0 ∈ {1, 2, . . .} such that Let N 0 = {n 0 , n 0 + 1, . . .} and n ∈ N 0 and transform equation (44) into thus, r 3 = u < r 3 , this is a contraction. From this claim and Lemma 3.2, we know that u = T n u has a positive fixed point in B 2 , denoted by u n . Then, we get has a positive ω-periodic solution u n with u n ≤ r 3 . Next, we prove that u n (t) have a uniform positive lower bound θ > 0 such that min t∈[0,ω] u n (t) ≥ θ for all n ∈ N 0 . The following formula is true, In the end, we need to illustrate the solution u n of equation (45) is that of the original equation (4), for this purpose, we prove {u ni } i∈N0 and {u ni } i∈N0 are compact.
Substituting equation (47) into equation (46), we have u n ≤ p ω u n + q ωr 3 |µ| + q ω |µ| for all (t, u) ∈ [0, ω] × (0, r 3 ]. We see from the above equation that u n is bounded if p ω < 1, i.e., In consequence, sequences {u n } n∈N0 and {u n } n∈N0 are bounded and equi-continuous family in C 1 ω := {u ∈ C(R, R) : u(t + ω) ≡ u(t), u (t + ω) ≡ u (t) for t ∈ R}. Using the Arzela-Ascoli theorem, it is clear that {u n } n∈N0 has a subsequence {u ni } i∈N0 , converging uniformly on R to a function u ∈ X. It follows from u n ≤ r 3 and u n (t) ≥ θ that u satisfies θ ≤ u(t) ≤ r 3 for all t ∈ R. Moreover, u ni satisfies the integral equation Letting i → ∞, we get Therefore, u is a positive ω-periodic solution for equation (4) and satisfies u ∈ (0, r 3 ]. Furthermore, we can show u < r 3 by noting u = r 3 , the argument similar to the proof of the first claim and we can yield a contradiction.
Remark 6. Comparing Theorem 2.3 with Theorem 3.5, it is clear that Theorem 3.5 embraces both a strong singularity and a weak singularity, while Theorem 2.3 just can be used to consider a strong singularity.

4.
Existence results (III). In the section, we consider the existence of positive ω-periodic solutions for equation (4) by the following fixed point theorem in cones, which can be found in [10].
In the following, we study the existence of positive ω-periodic solutions for equation (4) by different values of µ, and the division method is the same as section 3.

µ > 0.
Theorem 4.2. Assume that µ > 0 and condition (F 2 ) hold. Furthermore, suppose the following conditions hold: There exists a positive real constant r 4 such that (H 8 ) There exists a positive real constant R 1 such that Then equation (4) has at least one positive ω-periodic solution u(t) with u ∈ [r 4 , R 1 ].
Proof. It is easy to see that an ω-periodic solution of equation (4) is just a fixed point of the operator equation u = Q 1 u, where the expression of Q 1 is the same as T 1 .
Proof. It is easy to see that an ω-periodic solution of equation (4) is just a fixed point of the operator equation u = Q 2 u, where the expression of Q 2 is the same as T 1 . Define Ω 2 1 := {u ∈ X : u < r 5 }, Ω 2 2 := {u ∈ X : u < R 2 }.

µ < −1.
Theorem 4.4. Assume that µ < −1 and condition (F 1 ) hold. Furthermore, suppose the following conditions hold: (H 11 ) There exists a positive real constant r 6 such that (H 12 ) There exists a positive real constant R 3 such that Then equation (4) has at least one positive ω-periodic solution u(t) with u ∈ [r 6 , R 3 ].
Proof. It is easy to see that an ω-periodic solution of equation (4) is just a fixed point of the operator equation u = Q 3 u, where the expression of Q 3 is the same as T 1 . Define sets Ω 3 1 := {u ∈ X : u < r 6 }, Ω 3 2 := {u ∈ X : u < R 3 }.