On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments
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Bibliographic record
Abstract
By using Krasnoselskii’s fixed point theorem, we prove that the following periodic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n minus"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n-</mml:annotation> </mml:semantics> </mml:math> </inline-formula> species Lotka-Volterra competition system with multiple deviating arguments <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis ModifyingAbove x With dot Subscript i Baseline left-parenthesis t right-parenthesis equals x Subscript i Baseline left-parenthesis t right-parenthesis left-bracket r Subscript i Baseline left-parenthesis t right-parenthesis minus sigma-summation Underscript j equals 1 Overscript n Endscripts a Subscript i j Baseline left-parenthesis t right-parenthesis x Subscript j Baseline left-parenthesis t minus tau Subscript i j Baseline left-parenthesis t right-parenthesis right-parenthesis right-bracket comma i equals 1 comma 2 comma ellipsis comma n comma"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ∗ </mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="1em"/> <mml:mspace width="1em"/> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>x</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:munderover> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo> − </mml:mo> <mml:msub> <mml:mi> τ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="2em"/> <mml:mspace width="1em"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} (\ast )\quad \quad \dot {x}_i(t)=x_i(t)\left [r_i(t)-\sum _{j=1}^{n}a_{ij}(t)x_j(t-\tau _{ij}(t)) \right ],\quad i=1, 2, \ldots , n,\qquad \quad \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> has at least one positive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega minus"> <mml:semantics> <mml:mrow> <mml:mi> ω </mml:mi> <mml:mo> − </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\omega -</mml:annotation> </mml:semantics> </mml:math> </inline-formula> periodic solution provided that the corresponding system of linear equations <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk asterisk right-parenthesis sigma-summation Underscript j equals 1 Overscript n Endscripts a overbar Subscript i j Baseline x Subscript j Baseline equals r overbar Subscript i Baseline comma i equals 1 comma 2 comma ellipsis comma n comma"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ∗ </mml:mo> <mml:mo> ∗ </mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="2em"/> <mml:mspace width="2em"/> <mml:mspace width="2em"/> <mml:mspace width="2em"/> <mml:mspace width="1em"/> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-Te
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it