In this paper, we prove and discuss averaging results for ordinary differential equations perturbed by a small parameter. The conditions we assume on the right-hand sides of the equations under which our averaging results are stated are more general than those considered in the literature. Indeed, often it is assumed that the right-hand sides of the equations are uniformly bounded and a Lipschitz condition is imposed on them. Sometimes this last condition is relaxed to the uniform continuity in the second variable uniformly with respect to the first one. In our results, we assume only that the right-hand sides of the equations are bounded by some locally Lebesgue integrable functions with the property that their indefinite integrals satisfy a Lipschitz-type condition. Also, we consider that they are only continuous in the second variable uniformly with respect to the first one.
The polyopisthocotylean Sparicotyle chrysophrii (Van Beneden et Hesse, 1863) was experimentally transmitted to gilthead seabream (Sparus aurata L.) by exposure to eggs (EGT) and by cohabitation with naturally parasitized fish (CT). In EGT trials, the infection was successfully transmitted by introducing containers with monogenean eggs in the fish tanks, with the highest infection level (85.7% prevalence, 3.3 mean intensity) achieved at 6 weeks post exposure (p.e.) to the infection dose of 650 eggs per tank. In CT trials, the progression of the infection was faster and reached higher levels than in EGT. When using small fish juveniles (30 g) (CT-2), infection reached 100% prevalence (mean intensity 8 monogeneans/fish) at 5 weeks p.e., but no eggs could be found in the fish even 10 weeks p.e. By contrast, when larger juveniles (150 g) were used (CT-1), infection levels were lower, but mature adults with eggs were detected starting from 8 weeks p.e. The effect of the parasite on the condition factor, haematocrit, haemoglobin concentration (Hb), red blood cell counts, mean corpuscular haemoglobin concentration (MCHC), mean corpuscular haemoglobin content (MCH) and mean cellular volume (MCV) of infected fish was studied in CT trials. The infection produced hypochromic anaemia, since Hb concentration significantly decreased at 5 and 10 weeks p.e. in CT-2 and at 8 weeks p.e. in CT-1. MCHC was significantly lower in parasitized than in control fish at 5 and 8 weeks p.e. in CT-2 and CT-1, respectively. Also in CT-1, MCH was lower and circulating immature erythrocytes, granulocytes and plasma cells were higher in infected fish than in control ones at 8 weeks p.e. The histopathological effects of the monogenean on the gills of naturally infected fish consisted of lamellar shortening, clubbing and synechiae. The proliferation of the epithelial tissue produced fusion of secondary lamellae, and abundant chloride cells were observed.