In a heavy Dutch oven, heat oil over medium high heat. Add alligator and sauté 20 minutes to render juices. Add onions, celery, bell pepper, garlic, and jalapeños. Sauté until vegetables are wilted, approximately 3 to 5 minutes. Add pinto beans, tomato sauce and chicken stock, bring to a low boil, and reduce to simmer. Add chili powder and cumin, stir well into mixture, and allow to cook one hour, stirring occasionally. Once alligator is tender, season to taste using salt and black pepper. This dish is always served at hunting camp dinners over spaghetti.