This theorem is similar to the existence

Lemma 2, Lemma 3 and Lemma 4 are formulated to test condition 4) of the Fishing principle in system (2), (3).
Lemma 2.
(See [2], [18, p. 126].) Suppose that  a=0a=0andequation(12)3(σ−1)<2(2σ−1)2L+1,whereL=−σ+12+(σ+1)24+σ(r−1).Then for system (2)condition 4) is satisfied.
Lemma 3.
(See [15, pp. 276, 270–271], [19, pp. 374–375, 351–352, 367–369].) Suppose that for system (2)with parameters (3)the inequalityequation(13)R<4(σ+1)1+1+8a0(σ+1)is valid.Then for system (2)the condition of Theorem 2is satisfied.
In [15] and [19] it VX222 is proved that condition (5) is satisfied for S=IS=I and ?(X,s)≡0?(X,s)≡0.
Here R(s)≡RR(s)≡R, σ(s)=σσ(s)=σ, a(s)∈[0,a0]a(s)∈[0,a0], a(0)=0a(0)=0, a(1)=a0a(1)=a0.
Lemma 4.
If conditions (12)and (13)are valid, then for system (2)condition 4) is satisfied.
Suppose, condition 4) is not true. Then for a certain s1∈[0,1]s1∈[0,1] such that a(s1)∈[0,a0]a(s1)∈[0,a0] condition 3) is valid. In death rate case, due to Theorem 1 one obtains that there exists s0∈[0,s1]s0∈[0,s1] such that for a(s0),σ,Ra(s0),σ,R there exists a homoclinic trajectory of the saddle x=y=r=0x=y=r=0. However, this contradicts to the assertion of Theorem 2 and thereby proves the lemma.  □