# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. (English) Zbl 1011.34031

The authors study the nonlinear impulse differential equations in the plane

$\begin{array}{cc}& {\stackrel{˙}{x}}_{1}={F}_{1}\left({x}_{1},{x}_{2}\right),\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{2}={F}_{2}\left({x}_{1},{x}_{2}\right),\hfill \\ & {x}_{1}\left({t}_{i}^{+}\right)={\theta }_{1}\left({x}_{1}\left({t}_{i}\right),{x}_{2}\left({t}_{i}\right)\right),\phantom{\rule{1.em}{0ex}}{x}_{2}\left({t}_{i}^{+}\right)={\theta }_{2}\left({x}_{1}\left({t}_{i}\right),{x}_{2}\left({t}_{i}\right)\right),\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

with ${t}_{i+1}-{t}_{i}=\tau >0$, $i=0,1,2,\cdots$, ${x}_{1},{x}_{2}\in ℝ$, ${\theta }_{1}$, ${\theta }_{2}$ two positive, suitably smooth functions of ${x}_{1}$: normal cell biomass and ${x}_{2}$: tumor cell biomass. They describe the competition between normal and tumor cells. Sufficient conditions for the existence of nontrivial, periodic solutions to (1) are given.

##### MSC:
 34C25 Periodic solutions of ODE 92C37 Cell biology 34A37 Differential equations with impulses 34D30 Structural stability of ODE and analogous concepts 92B05 General biology and biomathematics