zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Impulsive control strategies in biological control of pesticide. (English) Zbl 1100.92071

Summary: By presenting and analyzing a pest-predator model under insecticides used impulsively, two impulsive strategies in biological control are put forward. The first strategy: the pulse period is fixed, but the proportional constant E 1 changes, which represents the fraction of pests killed by applying insecticides. For this scheme, two thresholds, E 1 ** and E 1 * for E 1 are obtained. If E 1 E 1 * , both the pest and predator (natural enemies) populations go to extinction. If E 1 ** <E 1 <E 1 * , the pest population converges to the semi-trivial periodic solution while the predator population tends to zero. If E 1 is less than E 1 ** but even if close to E 1 ** , there exists a unique positive periodic solution via bifurcation, which implies both the pest and the predator populations oscillate with a positive amplitude. In this case, the pest population is killed to the maximum extent while the natural enemies are preserved to avoid extinction.

The second strategy: the proportional constant E 1 is fixed (E 1 <E 1 * firstly), but the pulse period changes. For this scheme, one threshold τ 0 for the pulse period τ is obtained. We can reach the same target as above by controlling the period impulsive effect τ<τ 0 , even if close to τ 0 . Our theoretical results are confirmed by numerical simulations.

93C15Control systems governed by ODE
93C95Applications of control theory
34A37Differential equations with impulses